PCM20210805_SICP_1.2.6_Example_Testing_for_Primality_I

### A Pluto.jl notebook ###
# v0.20.4

using Markdown
using InteractiveUtils

# ╔═╡ b58f3d47-bc97-4b54-ad3e-19b775e6849c
begin 
	using Pluto
	using Plots	
	using LaTeXStrings
	using GraphRecipes
	using Primes
	#----------------------------------------------------------------
	println("pkgversion(Pluto)         = ", pkgversion(Pluto))
	println("pkgversion(Plots)         = ", pkgversion(Plots))
	println("pkgversion(LaTeXStrings)  = ", pkgversion(LaTeXStrings))
	println("pkgversion(GraphRecipes)  = ", pkgversion(GraphRecipes))
	println("pkgversion(Primes)        = ", pkgversion(Primes))
end # begin

# ╔═╡ fcaf83d0-f611-11eb-1bc9-31384cf290cb
md"
==================================================================================
#### SICP: [1.2.6 Example: Testing for Primality I](https://mitp-content-server.mit.edu/books/content/sectbyfn/books_pres_0/6515/sicp.zip/full-text/book/book.html)

##### file: PCM20210805_SICP\_1.2.6\_Example\_Testing\_for\_Primality\_I

##### Julia/Pluto.jl-code (1.11.3/0.20.4) by PCM *** 2025/02/04 ***
==================================================================================
"

# ╔═╡ 6d09aba9-b862-4e38-a4a9-fd718d4744c4
md"
##### 0. Introduction
*Since ancient times, mathematicians have been fascinated by problems concerning prime numbers, and many people have worked on the problem of determining ways to test if numbers are prime.*(SICP, 1996, 2016)
"

# ╔═╡ 23534646-17d3-4cba-8505-10991158a47c
md"
---
##### 1. Topics
- *exponentiation* with ^
- *case* analysis by $if$ <condition> <epression> $elseif$ <condition> <epression> $else$ <epression> $end$
- function-wide *scope*
- *jumps* by $break$
- *Fermat*'s Little Theorem
- *coprimes*
- *package* $Primes$

"

# ╔═╡ b64b7617-c882-44b3-9400-863138f3678e
md"
---
##### 2. Libraries
"

# ╔═╡ 35a00708-a1ff-4b0c-a4eb-50452e394e06
md"
---
##### 3. SICP-Scheme-like *functional* Julia
###### 3.1 *Generic* (*Polymorphic*) function $isprime$: Searching for *Divisors* (= *Factors*)
*One way to test if a number is prime is to find the number's divisors. The following program finds the smallest integral divisor (greater than $1$) of a given number $n$. It does this in a straightforward way, by testing $n$ for divisibility by successive integers starting with* $2$.(SICP, 1996, 2016)

"

# ╔═╡ fc96aecc-a50e-4b17-aeda-1294bdf2ec0b
md"
---
###### Requirement: function $square$
"

# ╔═╡ 66a32388-59bf-4cda-946d-a299487c8760
^(3, 3)              # prefix exponential operator '^'; ==> 27 --> :)

# ╔═╡ da94c3aa-381d-4604-bdd8-298f74ccd998
square(x) = ^(x, 2)  # definition of square

# ╔═╡ 6dce0f0d-290f-429b-8c4d-d9a3e9ea5357
square(11), typeof(square(11))           # for Int64

# ╔═╡ 22f5327d-b4ea-4eae-badb-cf8a139b4bd3
square(11.), typeof(square(11.))         # for Float64

# ╔═╡ d123c9b7-7473-40b3-981c-03a4c88e2126
md"
---
###### Requirement: function $remainder$ ...
... *from Euclidean division, returning a value of the same sign as $x$, and smaller in magnitude than* $y$. (see Julia-doc) 
"

# ╔═╡ 414fada3-20bd-4085-9dcc-ae279600ad30
remainder = rem

# ╔═╡ 038c0c86-ec82-4f7d-a271-a06490995528
%(3, 9), rem(3, 9), remainder(3, 9)

# ╔═╡ 3107665f-2e0e-493e-aec8-e536ccea0492
%(9, 3), rem(9, 3), remainder(9, 3)

# ╔═╡ fc40132c-b9b4-4c04-a67a-88f382482189
%(13, 9), rem(13, 9), remainder(13, 9)

# ╔═╡ c14293eb-0a71-4e59-8a85-d65354ca1db0
%(9, 13), rem(9, 13), remainder(9, 13)

# ╔═╡ 9b30ad81-9829-48f6-b125-9a61ee3bcc05
%(13.0, 9.0), rem(13.0, 9.0), remainder(13.0, 9.0)

# ╔═╡ 4c9de0d0-dd9d-4371-9a1b-9523d641ad32
%(9.0, 13.0), rem(9.0, 13.0), remainder(9.0, 13.0)

# ╔═╡ d50c7a2f-a7cf-4b1a-8b07-c3cdf96024d0
md"
---
###### Requirement: function $divides$ ... 
"

# ╔═╡ 452f1c9a-d2b5-46bf-adcc-af9f80b62d68
divides(a, b) = ==(remainder(b, a), 0)

# ╔═╡ 19cf97ec-4c9a-4069-9dce-fa265bb904e5
divides(3, 9)

# ╔═╡ d56f43ca-11dc-491f-b6ad-f332a2e2731c
divides(3, 10)

# ╔═╡ c37fb2db-62ef-4eeb-a7cf-27a3cde6bb64
md"
---
###### Requirement: function $smallest\text{-}divisor$
(SICP, 1996; 2016, p.65f)
"

# ╔═╡ e78e17f2-c9fa-4f5b-aefc-8c3e580f117f
function find_divisor(n, test_divisor)
	>(square(test_divisor), n) ? n :            # stop condition
	divides(test_divisor, n) ? test_divisor :
		find_divisor(n, +(test_divisor, 1))     # tail recursive call
end # function find_divisor

# ╔═╡ afcd5e13-b61a-491a-bf4f-47ea59fe112d
smallest_divisor(n) = find_divisor(n, 2)

# ╔═╡ c1fd79b5-abfa-44b6-bc10-783f4b1c9924
smallest_divisor(1)             # ==> 1 --> :)

# ╔═╡ f1aa9700-1c4a-4a7d-8563-595984bd4a19
smallest_divisor(7)             # ==> 7 --> :), because 7 is prime

# ╔═╡ 5504eb77-0e64-4321-8db9-6449deaa401f
smallest_divisor(11)            # ==> 11 --> :), because 11 is prime

# ╔═╡ b8e74e16-deed-4903-aaea-d68e3e9d6024
smallest_divisor(15)            # ==> 3 --> :)

# ╔═╡ 46cc1617-f160-4041-bd79-193d311afb61
smallest_divisor(97)            # ==> 97 --> :), because 97 is prime

# ╔═╡ db02e8a1-4131-4c0f-9975-6223dafd1631
md"
---
###### *Generic* (*Polymorphic*) function $isprime$
(SICP, 1996; 2016, p.66)

The [first 25 primes](https://en.wikipedia.org/wiki/Prime_number) are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

"

# ╔═╡ 768cc486-5e6e-4ce1-bf2c-a35151a4793f
myIsprime(n) = ==(n, smallest_divisor(n)) 

# ╔═╡ dac9cfc1-6b2d-4458-82a6-cf9e2c79fcc1
myIsprime(1)                    # ==> true --> :); some say 1 is not a prime

# ╔═╡ b5b5a466-e78e-49c0-8ee9-765644596e27
myIsprime(2)                    # ==> true --> :); 2 is prime

# ╔═╡ 56beed19-5ee7-458b-bd50-3afb9e38e342
myIsprime(3)                    # ==> true --> :); 3 is prime

# ╔═╡ a344aaaa-af28-4345-8e96-a6b735fb6023
myIsprime(4)                    # ==> false --> :)

# ╔═╡ eb89783e-e948-42d8-b7d5-857a719da9b4
myIsprime(5)                    # ==> true --> :); 5 is prime

# ╔═╡ 2754573a-24ea-4a7b-85c3-303004aec2e2
myIsprime(6)                    # ==> false --> :)

# ╔═╡ 2a3be054-6361-4b44-9ceb-465547ce3803
myIsprime(97)                   # ==> true --> :); 97 is prime

# ╔═╡ 6bd55608-02f5-4b95-8ad9-597abdf106b6
md"
###### Sample from the [First 100,008 Primes](https://primes.utm.edu/lists/small/100000.txt)
"

# ╔═╡ 4d06eb30-b117-4b02-b04c-6c459a73bcd8
myIsprime(99989)                # ==> true --> :)

# ╔═╡ 33fb902c-bece-4876-a731-3eaf7ffe2ff4
myIsprime(99991)                # ==> true --> :)

# ╔═╡ 9bc0eb42-79b0-4d7b-b472-07abae671439
myIsprime(99993)                # ==> false --> :); because 3 * 33331 = 99993

# ╔═╡ 57187f78-2aeb-4284-90da-b529a298eb91
myIsprime(100003)               # ==> true --> :)

# ╔═╡ 8d69e081-99d6-404a-805a-5d12821c5f5a
md"
###### [Carmichael numbers](https://en.wikipedia.org/wiki/Carmichael_number) (= [Fermat *pseudo*primes](https://en.wikipedia.org/wiki/Fermat_pseudoprime))
A *Carmichael number* will *usually* pass *Fermat primality test* even though it is *not* actually prime. But this test $myIsprime$ is searching divisors *exhausively*, so its perfect but not effective. *...there are $20.138.200$ Carmichael numbers between $1$ and $10^{21}$*. [(*Wikipedia*)](https://en.wikipedia.org/wiki/Carmichael_number)

The first *seven* Carmichael numbers *were all found by the Czech mathematician Václav Šimerka in 1885* ([*Wikipedia*](https://en.wikipedia.org/wiki/Carmichael_number)):
$561, 1105, 1729, 2465, 2821, 6601, 8911$
"

# ╔═╡ 6a7a2391-9cbc-44ef-83f8-5252c1454ac2
md"
###### 1st Carmichael number
"

# ╔═╡ fa620300-3307-4bda-97de-1c49fbf58d61
myIsprime(561)                  # ==> false --> :)

# ╔═╡ b5278a66-dbca-44d0-96ad-bd8e24e90ee3
md"
###### 2nd Carmichael number
"

# ╔═╡ 81147945-ae09-4fa3-a9ef-288a8c42744e
myIsprime(1105)                 # ==> false --> :)

# ╔═╡ e555ade7-e8f8-4158-8b68-0af132d8861a
md"
###### 3rd Carmichael number
"

# ╔═╡ a37e9857-fc51-4fdc-b8a5-50f6607f52bf
myIsprime(1729)                 # ==> false --> :)

# ╔═╡ d641ab56-f97a-4cab-820d-2d1ce7b3c546
md"
###### 4th Carmichael number
"

# ╔═╡ 04a9798b-fc72-455e-be71-62842167b292
myIsprime(2465)                 # ==> false --> :)

# ╔═╡ 1c6fbf70-c098-44cc-ac84-28acc220da7d
md"
###### 5th Carmichael number
"

# ╔═╡ af7c8d34-5bf3-4e5a-851e-4b7d13b5f6b4
myIsprime(2821)                 # ==> false --> :)

# ╔═╡ 8cec6bc7-37fd-43b0-81de-863d6a93e153
md"
###### 6th Carmichael number
"

# ╔═╡ 28eca35a-ff7a-4f07-84dd-08d8329f8fd2
myIsprime(6601)                 # ==> false --> :)

# ╔═╡ fbc3c07c-7db4-4c69-add6-5135d33c2ebd
md"
###### 7th Carmichael number
"

# ╔═╡ eb73d8b4-c26f-49a7-9790-49c72fbf85e9
myIsprime(8911)                 # ==> false --> :)

# ╔═╡ 113f62c9-e1b7-44f9-a113-d90be226ef51
md"
###### 8th Carmichael number
"

# ╔═╡ 25dad7c6-453d-4849-aeb1-cc56ac1c6faa
myIsprime(10585)                # ==> false --> :)

# ╔═╡ d6bc8877-7093-4c3c-a256-ab1409183866
md"
###### 9th Carmichael number
"

# ╔═╡ 3e9d4b35-0e9d-4a23-874c-82ed4edee8fb
myIsprime(15841)                # ==> false --> :)

# ╔═╡ be8ba70d-fd85-40d9-bd91-79907b148439
md"
###### 10th Carmichael number
"

# ╔═╡ 26c58152-1e0a-45e4-bbf9-effefff6c695
myIsprime(29341)                # ==> false --> :)

# ╔═╡ 7d4832ca-cd8c-49af-aa3e-df0104dc677c
md"
###### 11th Carmichael number
"

# ╔═╡ ac8e5704-a619-4545-95c3-313f4e8d6585
myIsprime(41041)                # ==> false --> :)

# ╔═╡ 714d49ef-ede2-439a-acac-d8065ca22771
md"
###### 12th Carmichael number
"

# ╔═╡ 6811e773-beb5-442a-88b6-61de219eb71d
myIsprime(46657)                # ==> false --> :)

# ╔═╡ 534db6f2-c7ea-4e67-9a29-66a1063a3722
md"
###### 13th Carmichael number
"

# ╔═╡ 6e8b2888-2c41-4868-a451-0f68a0256056
myIsprime(52633)                # ==> false --> :)

# ╔═╡ 376f22c8-6323-4f4f-ba68-d4066b0c08f5
md"
###### 14th Carmichael number
"

# ╔═╡ b9994c20-d8b9-43e0-839f-0727c2280868
myIsprime(62745)                # ==> false --> :)

# ╔═╡ 38ac1686-0f3a-4969-907d-16e9ab98ea65
md"
###### 15th Carmichael number
"

# ╔═╡ dcc6e279-3775-404e-adec-23fe32388032
myIsprime(63973)                # ==> false --> :)

# ╔═╡ 42f795ad-acd6-49a1-8328-61217325c991
md"
###### 16th Carmichael number
"

# ╔═╡ 148f7d98-dbd5-473c-ab4a-6fa0559f8413
myIsprime(75361)                # ==> false --> :)

# ╔═╡ e698ce81-6cd0-4a9a-931a-5994bf91dc39
md"
---
###### 3.2 Variation $isprime2$ of $isprime$: *Generic* (*Polymorphic*) function 
with *local* function definitions
"

# ╔═╡ 9979c1e4-349b-4e0a-87a0-39de45ecdf4c
function myIsprime2(n) 
	#-------------------------------------------
	remainder = %          # idiomatic Julia '%'
	#-------------------------------------------
	square(x) = ^(x, 2)    # infix operator '^'
	#-------------------------------------------
	divides(a, b) = ==(remainder(b, a), 0)
	#-------------------------------------------
	smallest_divisor() = find_divisor(2)
	#-----------------------------------------------------------------------------
	function find_divisor(test_divisor)  
		>(square(test_divisor), n) ? n : 
		divides(test_divisor, n) ? test_divisor : 
			find_divisor( +(test_divisor, 1))
	end # function find_divisor
	#-----------------------------------------------------------------------------
	n == smallest_divisor() 
end # function myIsprime2

# ╔═╡ e6ac837a-adc0-4114-9b09-edb43ab4d5bb
myIsprime2(1)                         # ==> true --> :); some say 1 is not a prime

# ╔═╡ c5ed48aa-74b8-4a68-958f-10aca604ea5b
myIsprime2(2)                         # ==> true --> :)

# ╔═╡ dca62763-1e1d-48c8-97dc-bc47bca6f2d3
myIsprime2(3)                         # ==> true --> :)

# ╔═╡ 692e04a0-ea2c-4ce4-bee2-2d3d4736259b
myIsprime2(4)                         # ==> false --> :)

# ╔═╡ df838016-92fc-4255-be0a-85fb0990bce7
myIsprime2(5)                         # ==> true --> :)

# ╔═╡ cb61b117-245b-4845-808c-70deb40f7061
myIsprime2(6)                         # ==> false --> :)

# ╔═╡ f2741526-1906-4ea4-8800-17d27a74ed18
myIsprime2(97)                        # ==> true --> :)

# ╔═╡ 218cc9d7-3c37-44a6-82b8-fecdf8e6fc02
myIsprime2(99989)                     # ==> true --> :)

# ╔═╡ d803272a-0090-495d-93a2-d56d2d9b6858
myIsprime2(99991)                     # ==> true --> :)

# ╔═╡ 23cdee96-73a6-4bdc-ba7d-627f84821ea5
myIsprime2(99993)                     # ==> false --> :); because 3 * 33331 = 99993

# ╔═╡ 01ba9d22-919c-4887-919e-88dcfbc7a03c
myIsprime2(100003)                    # ==> true --> :)

# ╔═╡ 4cb5f5b4-c0a5-4bac-93e8-1b5069e3863a
md"
###### [Carmichael numbers](https://en.wikipedia.org/wiki/Carmichael_number) (= [Fermat *pseudo*primes](https://en.wikipedia.org/wiki/Fermat_pseudoprime))
A *Carmichael number* will *usually* pass *Fermat primality test* even though it is *not* actually prime. But this test $myIsprime2$ is searching divisors *exhausively*, so its perfect but not effective.
"

# ╔═╡ c9f0881e-95d4-44ab-8a1b-84b9b1636478
md"
###### 1st Carmichael number
"

# ╔═╡ 894d5371-66dd-4cfa-83ad-a1a5ef96fe81
myIsprime2(561)                       # ==> false --> :)

# ╔═╡ 9ccc6095-fb94-4bbe-8b79-adf38cacadcf
md"
###### 2nd Carmichael number
"

# ╔═╡ 000041cd-e527-46ba-bdbb-c899ccc0567d
myIsprime2(1105)                      # ==> false --> :)

# ╔═╡ d90c320e-c578-45ea-bc98-f54b63a83a4b
md"
###### 3rd Carmichael number
"

# ╔═╡ b7f68f0f-6dc4-4131-947e-ec88ad3827b3
myIsprime2(1729)                      # ==> false --> :)

# ╔═╡ 0ed8c711-5530-454f-8c7f-8f72fe90032b
md"
###### 4th Carmichael number
"

# ╔═╡ bfaca850-d65f-4354-9149-1d97f23cad8b
myIsprime2(2465)                      # ==> false --> :)

# ╔═╡ b9fee51c-774b-474d-a508-962df966b5b0
md"
###### 5th Carmichael number
"

# ╔═╡ 7906ee7f-c102-42c6-beff-aa08d6f2ebdf
myIsprime2(2821)                      # ==> false --> :)

# ╔═╡ 977e806a-b361-4d54-bf12-ea7e46e64bc6
md"
###### 6th Carmichael number
"

# ╔═╡ 5eed380c-d7f3-4a64-ab58-b2598e4925e1
myIsprime2(6601)                      # ==> false --> :)

# ╔═╡ f0189ee0-ee8f-450f-a909-73050adfb965
md"
###### 7th Carmichael number
"

# ╔═╡ c7159488-3e73-4327-bd66-0f80cdc5937b
myIsprime2(8911)                      # ==> false --> :)

# ╔═╡ c6a5e5cb-c266-4315-88a6-e02990d7e21c
md"
###### 8th Carmichael number
"

# ╔═╡ 29d544ec-ca31-403e-bb47-ac2cb75bb211
myIsprime2(10585)                     # ==> false --> :)

# ╔═╡ 2febaefa-c436-4b93-b312-30c2dfda4f9a
md"
###### 9th Carmichael number
"

# ╔═╡ a2800255-5583-448f-9b89-8ec4b1bdfe47
myIsprime2(15841)                     # ==> false --> :)

# ╔═╡ 1624efd8-abb1-47b4-b42d-b190c0a93099
md"
###### 10th Carmichael number
"

# ╔═╡ 7297f795-c99d-46f0-bf6b-30ba3954b4d4
myIsprime2(29341)                     # ==> false --> :)

# ╔═╡ 8edd5779-2050-4255-aee0-1b28edff2540
md"
###### 11th Carmichael number
"

# ╔═╡ b1cc4859-166c-4836-817f-6bfe480978ca
myIsprime2(41041)                     # ==> false --> :)

# ╔═╡ ec478f2e-1b78-44f6-a99f-7822fb1594b8
md"
###### 12th Carmichael number
"

# ╔═╡ 186e941a-d81b-4f64-a974-346268d74654
myIsprime2(46657)                     # ==> false --> :)

# ╔═╡ e5fa1992-0f35-467f-a113-66a0f6a45b48
md"
###### 13th Carmichael number
"

# ╔═╡ f7288e68-eb85-4a4a-b744-d7fb82446d17
myIsprime2(52633)                     # ==> false --> :)

# ╔═╡ b9732f72-9344-45f4-9b44-09ff50c1c373
md"
###### 14th Carmichael number
"

# ╔═╡ 0e33cbc9-159f-4d26-b945-490e3922037a
myIsprime2(62745)                     # ==> false --> :)

# ╔═╡ f150a27b-728c-47ef-945d-62331de9f474
md"
###### 15th Carmichael number
"

# ╔═╡ ecb30694-9742-4519-8b51-c1d520a7335e
myIsprime2(63973)                     # ==> false --> :)

# ╔═╡ 6cece608-0153-4415-963f-e94bc54c32c2
md"
###### 16th Carmichael number
"

# ╔═╡ 134fb4b5-29d2-41be-8ccd-5dff7a7d6cf0
myIsprime2(75361)                     # ==> false --> :)

# ╔═╡ 374eccff-3888-4760-9ce7-09b388beaed0
md"
---
###### 3.3 [Fermat's Little Theorem](https://en.wikipedia.org/wiki/Fermat%27s_little_theorem) (SICP version)

*If $n$ is a prime number and $a$ is any positive integer less than $n$, then $a$ raised to the $n$th power is congruent to $a$ modulo $n$* (SICP, 1996; [2016, p.67](https://web.mit.edu/6.001/6.037/sicp.pdf); Cormen et. al., 2022, p.932, 944):

$\boxed{\text{If } isprime(n) \text{ then } \forall a \in \{a|1 < a < n\}: a^n \equiv a\ (\bmod n)}$

where $a$ is called *basis* and where:

$a^n \equiv a\ (\bmod n) \text{ is an abbreviation for: } a^n\ (\bmod n) = a\ (\bmod n).$

Below is another (shorter) version of the theorem, which is *almost perfect* (Cormen et al., 2022, p.944). 

"

# ╔═╡ 84499038-4ed6-46fc-a6d1-70a5964c3f3e
md"
---
###### 3.3.1 *Modular-Exponentiation* $expmod1$
*To implement the Fermat test, we need a procedure that computes
the exponential of a number modulo another number* (SICP, 1996; 2016, p.67; Cormen et al., 2022, p.934):

$a^b\ (\bmod n) = \cases{1   \;\;\;\;\;\;\;\;\;\;\,\text{ if }\ b = 0 \\(a^{b/2})^2  \;\;\text{ if } (b > 0) \land iseven(b) \\ a \cdot a^{b-1} \;\text{ if }(b > 0) \land isodd(b)}$
"

# ╔═╡ 99f32ce2-e7c5-4a0a-b466-d7ecf5ef6af4
md"
... with *self-defined* type FloatOrSigned
"

# ╔═╡ 8bbdd3d4-9a2b-478d-8ba4-7ca3a10aa86e
FloatOrInteger = Union{AbstractFloat, Signed}

# ╔═╡ 0af40eda-a6fd-4fb3-9d77-129c02d66b79
# a^b mod n or baseA^expB mod n
#idiomatic Julia-code '%', '^2'
function expmod1(baseA::Signed, expB::FloatOrInteger, modN::Signed)
	#---------------------------------------------------------
	square(x) = x^2
	#---------------------------------------------------------
	remainder = %
	#---------------------------------------------------------
	even(n)::Bool = remainder(n, 2) == 0
	#---------------------------------------------------------
	if expB == 0
		1
	elseif even(expB)
		remainder(square(expmod1(baseA, expB/2, modN)), modN)
	else
		remainder(baseA * expmod1(baseA, expB-1, modN), modN)
	end # if
end # function expmod1

# ╔═╡ bff175b8-7637-40ae-84a5-3047a5c71d30
md"
$2^3\ (\bmod 3)= 8\ (\bmod 3) = 2$
"

# ╔═╡ 5ee7549a-dff7-4809-b63c-ebf572ad10eb
expmod1(2, 3, 3)

# ╔═╡ 65c473e2-c57a-4045-bd41-1f0ee046ef0f
md"
$2^5\ (\bmod 5)= 32\ (\bmod 5) = 2$
"

# ╔═╡ 770100ce-9cd8-4892-a3c0-d15c97192b5b
expmod1(2, 5, 5)

# ╔═╡ d83d7003-094c-4e99-8762-d642156581d0
md"
$3^5\ (\bmod 5)= 243\ (\bmod 5) = 3$
"

# ╔═╡ 247589bd-70b9-4677-bf8f-472aa4d57804
expmod1(3, 5, 5)

# ╔═╡ 24479ece-4777-4c18-a846-682c2364486d
md"
---
###### 3.3.2 *Modular-Exponentiation* $expmod2$
"

# ╔═╡ 708d6883-e125-489e-bbe7-76ff0fa59aa6
#idiomatic Julia-code '%', '^2', 'convert'
function expmod2(baseA::Signed, expB::FloatOrInteger, modN::Signed)::Signed
	even(x) = x%2 == 0
	if expB == 0
		1
	elseif even(expB)
		%(expmod2(baseA, expB/2, modN)^2, modN)
	else
		%(baseA * expmod2(baseA, expB-1, modN), modN)
	end # if
end # function expmod2

# ╔═╡ 80174e42-0389-4efe-b80a-97625860c4dd
expmod2(2, 3, 3)                   # ==> 2 --> :)

# ╔═╡ 18bcfbd7-be4e-45d3-8e80-0953a422368d
expmod1(2, 5, 5)                   # ==> 2 --> :)

# ╔═╡ d3002fe5-3fc9-493e-93ca-e0e243845d68
expmod2(3, 5, 5)                   # ==> 3 --> :)

# ╔═╡ cebc2513-0f94-41df-ab52-9bfe6815bb8d
md"
---
###### 3.3.3 *Fermat Test* (SICP, 1996; 2016, p.68) $fermat\_test1$
*The Fermat test is performed by choosing at random a number a between 1 and n - 1 inclusive and checking whether the remainder modulo n of the nth power of a is equal to a. The random number a is chosen using the procedure random, which we assume is included as a primitive in Scheme. Random returns a nonnegative integer less than its integer input. Hence, to obtain a random number between 1 and n - 1, we call random with an input of n - 1 and add 1 to the result:* (SICP, 1996; 2016, p.68)
"

# ╔═╡ e57b3eb9-a394-4253-8f23-76ff51bc1998
histogram([rand(1:10) for i in 1:10E2], bins=1:11)                

# ╔═╡ 8d905cbe-ca91-413c-ae68-0c7613685b5b
rand(1:3)

# ╔═╡ 7079c8a5-af5b-4083-b494-aaaf0ed1336a
# idiomatic Julia-code with 'rand'
function fermat_test1(n::Signed)::Bool 
	#--------------------------------------------------------------------
	random(n) = rand(1:n) - 1 # MIT-Scheme: 0 ≤ random(n) ≤ n-1
	#--------------------------------------------------------------------
	# try_it(a) = expmod2(a, n, n) == a  # short form as in SICP
	try_it(a) = expmod2(a, n, n) == mod(a, n) # long form as in theorem
	#--------------------------------------------------------------------
	try_it(1 + random(n-1))
end # fermat_test1

# ╔═╡ cd1ed62c-086a-4d92-b625-dfce3362f1c6
fermat_test1(2)             # ==> true --> :)

# ╔═╡ f3dc6cb1-d360-4684-a860-09da9180456a
fermat_test1(3)             # ==> true --> :)

# ╔═╡ f22888d3-26a2-4dc5-af18-4f9e5e05ba91
fermat_test1(4)             # ==> false --> :) ; switches between true / false

# ╔═╡ e7b85f55-93e2-41d5-8a0c-2948e943c3f2
fermat_test1(5)             # ==> true --> :)

# ╔═╡ ac859543-dae4-49de-9afc-ea9c5d6600e8
fermat_test1(6)             # ==> false --> :) ; switches between true / false

# ╔═╡ 49ad573f-078e-4ebb-a1e2-da27e6c46a1d
fermat_test1(7)             # ==> true --> :)

# ╔═╡ 643e4865-a682-4ef5-9c04-1e058d60f01b
fermat_test1(9)             # ==> false --> :)

# ╔═╡ 6d344ac0-c073-45ae-8d74-532ae7001a92
fermat_test1(97)            # ==> true --> :)

# ╔═╡ c9403479-b7e2-4391-a058-541501434de0
md"
---
###### 3.3.4 *Fermat Test* with *Replications*: $fast\_prime1$ 
*The following procedure runs the test a given number of times, as specified by a parameter. Its value is true if the test succeeds every time, and false otherwise.* (SICP, 1996; 2016, p.68)
"

# ╔═╡ 8a3e42bf-6c51-4201-b894-a96bc3b8c6b6
function fast_prime1(n::Signed, times::Signed)::Bool
	if 	times == 0
		true
	elseif fermat_test1(n)
		fast_prime1(n, times-1)
	else 
		false
	end # if
end # fast_prime1

# ╔═╡ cbdc8f90-774b-49ad-8736-033e4d98fbd3
fast_prime1(2, 10)          # ==> true --> :)

# ╔═╡ ce83ac3a-974f-4c2a-b407-16f89055c66c
fast_prime1(3, 10)          # ==> true --> :)

# ╔═╡ b90b682f-e083-44b9-93e8-2d7f143c8386
fast_prime1(4, 10)          # ==> false --> :)

# ╔═╡ ff66c235-6d17-42ad-bbfd-7b4cd5dd0bac
fast_prime1(4, 100)         # ==> false --> :)

# ╔═╡ a8d176f8-af2e-4fbe-a4ae-2588443f80e1
fast_prime1(5, 10)          # ==> true --> :)

# ╔═╡ 2408748e-f577-48b4-8828-e8bc59914a7a
fast_prime1(6, 10)          # ==> false --> :)

# ╔═╡ 13d75f8a-edc4-4042-8473-7eb24764b404
fast_prime1(7, 10)          # ==> true --> :)

# ╔═╡ 5474b2e3-ca51-42ce-9a61-f8a22420a162
fast_prime1(9, 19)          # ==> false --> :)

# ╔═╡ e2116f5a-636b-4194-9e5f-acffa2656a1c
fast_prime1(11, 10)         # ==> true --> :)

# ╔═╡ b2432745-1d5e-4f47-8c8e-dfa360bf3923
fast_prime1(97, 10)         # ==> true --> :)

# ╔═╡ 4ca1868f-c694-47fe-be38-6f1899c8e902
fast_prime1(99989, 10)      # ==> true --> :)

# ╔═╡ 41e11d01-7a92-4663-8b79-3d7302cbc985
fast_prime1(99991, 10)      # ==> true --> :)

# ╔═╡ 2275dc7a-43d9-47aa-a5ab-e1b53a8c7df2
fast_prime1(99993, 10)      # ==> false --> :); 99993 = 3*33331

# ╔═╡ 442e1767-4055-4a49-9f46-665240a0537d
fast_prime1(100003, 10)     # ==> true --> :)

# ╔═╡ cac5241d-8e7c-4c78-87ff-9df4d12632af
md"
###### Carmichael numbers fool Fermat's Test
"

# ╔═╡ 8cbdd3f2-e7d5-4e78-9d0b-34b43ac14b40
md"
###### 1st Carmichael number
"

# ╔═╡ eb712016-f15a-43aa-96b0-e129bff835f8
fast_prime1(561, 100)        # ==> true --> :( ; test was fooled

# ╔═╡ f7513770-7b6c-4bc8-b9fb-ca306f0cb9d8
md"
###### 2nd Carmichael number
"

# ╔═╡ f05ef05e-f1be-4706-b950-77e273c7884e
fast_prime1(1105, 100)       # ==> true --> :( ; test was fooled

# ╔═╡ c433c8bd-43e4-463a-8ff2-7b6bb70aaef3
md"
###### 3rd Carmichael number
"

# ╔═╡ 681da4df-ecb6-40ec-a5b8-03d49b25822c
fast_prime1(1729, 100)       # ==> true --> :( ; test was fooled

# ╔═╡ 5bcdaf47-9be0-4824-bd99-da274e30a360
md"
###### 4th Carmichael number
"

# ╔═╡ cb5d584d-5928-4223-9daf-139397b2e380
fast_prime1(2465, 100)       # ==> true --> :( ; test was fooled

# ╔═╡ 72c045cb-9655-4f2c-8340-4d9b19e11fce
md"
###### 5th Carmichael number
"

# ╔═╡ 6882300e-3e56-4eac-b7de-b6dd78a4d5cb
fast_prime1(2821, 100)       # ==> true --> :( ; test was fooled

# ╔═╡ 3351208c-94a4-4cc4-a3d0-851b0986010a
md"
###### 6th Carmichael number
"

# ╔═╡ 55293415-c1d3-4f8d-ab45-098e4a86d295
fast_prime1(6601, 100)       # ==> true --> :( ; test was fooled

# ╔═╡ 8408cbef-007b-4b64-a7d3-be4944113217
md"
###### 7th Carmichael number
"

# ╔═╡ 2a484f43-8e7b-4e6e-88c0-1d27a45cf4aa
fast_prime1(8911, 100)       # ==> true --> :( ; test was fooled

# ╔═╡ 076c461a-7a4b-42d7-a7fd-1006a8fa0f89
md"
###### 8th Carmichael number
"

# ╔═╡ 182fe756-65b6-43ee-8238-a8e6bd7950ee
fast_prime1(10585, 100)      # ==> true --> :( ; test was fooled

# ╔═╡ caef03c1-fe96-4504-a100-3478312e79c0
md"
###### 9th Carmichael number
"

# ╔═╡ bd896dca-d317-431a-89f3-a57dfef64f8f
fast_prime1(15841, 100)      # ==> true --> :( ; test was fooled

# ╔═╡ 742b6b98-d66b-41e5-96bf-8f3882047e6a
md"
###### 10th Carmichael number
"

# ╔═╡ ca245186-4ad0-4fd1-8cbb-7310f455b3cb
fast_prime1(29341, 100)      # ==> true --> :( ; test was fooled

# ╔═╡ 9d643e66-9504-4a40-81d3-690c2df98c5f
md"
###### 11th Carmichael number
"

# ╔═╡ 7aab20ba-4685-47fe-8337-1037c686909f
fast_prime1(41041, 100)      # ==> true --> :( ; test was fooled

# ╔═╡ 7477cc56-adfd-44bb-959b-819dc63fba9c
md"
###### 12th Carmichael number
"

# ╔═╡ 8b82b7b8-ba1f-4f7f-897e-c13f78cdf847
fast_prime1(46657, 100)      # ==> true --> :( ; test was fooled

# ╔═╡ 7c3b5bfb-fb6e-4d63-80f6-7a8a7dd1c0a6
md"
###### 13th Carmichael number
"

# ╔═╡ 8fcf9b3f-e7d3-4e33-b671-da484cfa07cc
fast_prime1(52633, 100)      # ==> true --> :( ; test was fooled

# ╔═╡ 96f3e1b1-442e-47ef-9306-b1debafb1e5c
md"
###### 14th Carmichael number
"

# ╔═╡ 1e6174e0-ed2e-4167-8a5f-0361b6b2ff1b
fast_prime1(62745, 100)      # ==> true --> :( ; test was fooled

# ╔═╡ 060e3768-239c-4e18-a8d9-bfb585e6c17e
md"
###### 15th Carmichael number
"

# ╔═╡ b0fe5cd7-e660-4dae-bd09-c826cfcd60f2
fast_prime1(63973, 100)      # ==> true --> :( ; test was fooled

# ╔═╡ 31b1939d-94bc-4c06-ace7-dbe70e2fb5cb
md"
###### 16th Carmichael number
"

# ╔═╡ a3febf5c-1b67-499a-96c3-77d94d7eaae2
fast_prime1(75361, 100)      # ==> true --> :( ; test was fooled

# ╔═╡ f3e9ad78-79f2-4aeb-8113-c8f50d05e68e
md"
---
###### 3.6 [Fermat's Little Theorem](https://en.wikipedia.org/wiki/Fermat%27s_little_theorem) (short version, e.g. Cormen et al., 2022, p.932)

If *basis* $a$ and *test candidate* $n$ are *coprime* then $gcd(a, n)==1$. In this case we can divide both sides of the equation 

$a^n\ (\bmod n) = a\ (\bmod n)$ 

by $a\ (\bmod n)$. Now, we get:

$a^{n-1}\ (\bmod n) = a^{1-1}\ (\bmod n) = a^0\ (\bmod n) = 1\ (\bmod n) = 1.$

So:

$a^{n-1}\ (\bmod n) = 1.$

We can include the condition $gcd(n, a) = 1$ as a precondition in *Fermat*'s test. If we *know* that $isprime(n)$ is *true* this check is redundant. In other cases we can use it as a *quick-check*. If $gcd(n, a) > 1$ then we know that $isprime(n)$ is *false* and $n$ is *composite*. Otherwise we are *uncertain* whether $n$ is *prime* or *pseudoprime* (=*Carmichael*) and we have to go on with testing.

Now we can write the *short* version of *Fermat's Little Theorem* as:

$\boxed{\text{If } isprime(n) \text{ then } \forall a \in \{a|(gcd(n, a) = 1) \land (1 < a < n) \}: a^{n-1}\  (\bmod n) = 1}$

When we abbreviate the conclusion $\text{ then } ... 1$ as:

$FTest(+|n, \forall a) =_{def} \forall a \in \{a|(gcd(n, a) = 1) \land (1 < a < n) \}: a^{n-1} (\bmod n) = 1.$

We can write the *short* version of *Fermat*'s Little Theorem as:

$\boxed{\text{If } isprime(n) \text{ then } FTest(+|n, \forall a)}.$

"

# ╔═╡ 88f02453-317c-4162-bb18-764715015a0b
# idiomatic Julia-code with 'rand'
function fermat_test2(n::Signed)::Bool 
	#--------------------------------------------------------------------
	random(n) = rand(1:n) - 1 # MIT-Scheme: 0 ≤ random(n) ≤ n-1
	#--------------------------------------------------------------------
	# try_it(a) = expmod2(a, n, n) == a  # short form as in SICP
	try_it(a) = (gcd(n, a) == 1) && (expmod2(a, n-1, n) == 1) 
	#--------------------------------------------------------------------
	try_it(1 + random(n-1))
end # fermat_test2

# ╔═╡ 5b9a1a53-2081-4778-aab6-a60fcfbef3b6
function fast_prime2(n::Signed, times::Signed)::Bool
	if 	times == 0
		true
	elseif fermat_test2(n)
		fast_prime2(n, times-1)
	else 
		false
	end # if
end # fast_prime2

# ╔═╡ d77a7757-c72d-4e8f-93c9-3009700ab975
fast_prime2(2, 100)           # ==> true --> :)

# ╔═╡ 595b9dda-541b-4732-a294-affce2b0c4e5
fast_prime2(3, 100)           # ==> true --> :)

# ╔═╡ c8d3613a-6d2b-4901-9321-aad8a99179cb
fast_prime2(4, 100)           # ==> false --> :)

# ╔═╡ fda84f34-1d43-446b-8c27-145b51812455
fast_prime2(5, 100)           # ==> true --> :)

# ╔═╡ 9a45cf8d-4d2b-48cc-9fc9-e0c4fd9468da
fast_prime2(6, 100)           # ==> false --> :)

# ╔═╡ 02659ecf-235a-4391-9998-167a69f6714e
fast_prime2(7, 100)           # ==> true --> :)

# ╔═╡ 7b7b0882-61be-4b76-96e8-18cb0d872ef6
fast_prime2(9, 100)           # ==> false --> :)

# ╔═╡ 2a6189a7-885d-42c1-bddb-c53891cfe972
fast_prime2(11, 100)          # ==> true --> :)

# ╔═╡ 0d3af18b-2cb0-4d25-b0b0-9df9f64caf9c
fast_prime2(97, 100)          # ==> true --> :)

# ╔═╡ 8e12127d-e814-4001-9100-77f679ed4c65
fast_prime2(99989, 100)       # ==> true --> :)

# ╔═╡ 293c9a6d-74cc-4e48-9386-4a3b946d5f20
fast_prime2(99991, 100)       # ==> true --> :)

# ╔═╡ 9a08b4be-03d3-4dee-aa7c-233a0427da15
fast_prime2(99993, 100)       # ==> false --> :), because 3*33331 = 99993

# ╔═╡ fa52d319-05b9-4798-9b9b-91d6ff907569
fast_prime2(100003, 100)      # ==> true --> :)

# ╔═╡ f64f11be-05fa-44a5-a14f-28044bac53aa
md"
###### Carmichael numbers *usually* fool Fermat's Test, but not $fast\_prime2$
"

# ╔═╡ 864506fb-9979-4cf9-a17d-fdfb0d88f5bf
md"
###### 1st Carmichael number
"

# ╔═╡ d021796a-f02e-42ab-911f-b018e13797b8
fast_prime2(561, 100)        # ==> false --> :) ; test could *not* be fooled

# ╔═╡ baf18a7f-7661-4fba-8cfa-5add5ea6d5e2
md"
###### 2nd Carmichael number
"

# ╔═╡ 53840c6f-68c9-42c3-b64f-123bce53b094
fast_prime2(1105, 100)       # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 42718dc3-6ccc-407f-a2c4-0163f4bcd422
md"
###### 3rd Carmichael number
"

# ╔═╡ 755421d0-9011-4e69-95c3-84f00e21b8ea
fast_prime2(1729, 100)       # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 9639eb4f-ae74-4310-8be5-cd8a546f9fea
md"
###### 4th Carmichael number
"

# ╔═╡ 7a05e505-cd9a-4719-9f04-ccbb48073847
fast_prime2(2465, 100)       # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 1d1da0d5-81ce-47dd-ad37-aa4156c02268
md"
###### 5th Carmichael number
"

# ╔═╡ 49802eb0-597c-4b67-bf32-a98bd21e3039
fast_prime2(2821, 100)       # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 32e1dc38-d2a6-4c6b-b817-fb958451b6a1
md"
###### 6th Carmichael number
"

# ╔═╡ f0f344a2-3a74-4bdb-a6a1-813376791e7e
fast_prime2(6601, 100)       # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 7c352835-0f7f-44ec-bbd4-2b8aee7d4fac
md"
###### 7th Carmichael number
"

# ╔═╡ 33f9961f-1236-465e-b182-13b4efdb5ab5
fast_prime2(8911, 100)       # ==> false --> :) ; test could *not* be fooled

# ╔═╡ b4b175bf-7515-471f-a649-d10763f0914e
md"
###### 8th Carmichael number
"

# ╔═╡ bfb82d1d-b568-417a-84fb-ce96978aaa3e
fast_prime2(10585, 100)      # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 1e0703f3-cdbf-4add-a853-986ae459844b
md"
###### 9th Carmichael number
"

# ╔═╡ b48b09c3-a076-419d-95e4-67446d6c9ac7
fast_prime2(15841, 100)      # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 1e3a90f8-a5fc-4ae3-aaae-5464d8fc23f3
md"
###### 10th Carmichael number
"

# ╔═╡ 223b1199-7cc0-4757-bbbc-5c5b02458c34
fast_prime2(29341, 100)      # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 3f71753e-39e3-4bf3-9c5b-0b3e142e8c8d
md"
###### 11th Carmichael number
"

# ╔═╡ bf1e0bb2-3a5c-4c4b-8ce3-e0823a44df07
fast_prime2(41041, 100)      # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 0eb07be2-3635-44f3-9a2a-4a51c0eb34cd
md"
###### 12th Carmichael number
"

# ╔═╡ fd4330e0-095c-4408-8617-7484c8e31b69
fast_prime2(46657, 100)      # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 0e429c35-a5d2-47a6-a42b-69c80926f867
md"
###### 13th Carmichael number
"

# ╔═╡ 2036003b-dfc1-421d-85c6-a3e3491e7b27
fast_prime2(52633, 100)      # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 95f53a85-dd50-4c15-9857-d1052fa00550
md"
###### 14th Carmichael number
"

# ╔═╡ 2c6cce53-42f4-407e-a5ab-531eb8098888
fast_prime2(62745, 100)      # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 070fee56-07a4-4f68-8020-e45834c9a98e
md"
###### 15th Carmichael number
"

# ╔═╡ 9d9d29fa-ce2e-4ab9-922d-94d513d274e1
fast_prime2(63973, 100)      # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 62199cc9-a4a6-4cf1-96c3-2594d891f392
md"
###### 16th Carmichael number
"

# ╔═╡ e53068ab-05d0-41d2-890d-47c4da7e3a79
fast_prime2(75361, 100)      # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 295d258d-3633-4c3f-acf4-f277679e4d2a
md"
---
##### 4. Idiomatic *imperative* Julia ...
... with types $Signed, Bool$ and  $%, while$, and $n^2$
"

# ╔═╡ 682dab37-9f5f-4f92-9795-c7d1c1d2179f
md"
###### 4.1 Iterative Function $isprime3$ 
with localized definitions and function-wide scope of $n$

"

# ╔═╡ 36ccf5f4-2106-4055-a255-457fe6e88185
# 'n' in local functions suppressed (but not in divides)
function myIsprime3(n::Signed)::Bool
	#-----------------------------------------------------------
	divides(a, b) = rem(b, a) == 0 
	#-----------------------------------------------------------
	smallest_divisor()::Signed = find_divisor(2)::Signed
	#-----------------------------------------------------------
	function find_divisor(test_divisor::Signed)::Signed  
	  while !((test_divisor^2 > n) || divides(test_divisor, n))
			test_divisor += 1
	  end # while
	  (test_divisor^2 > n) ? n : test_divisor
	end # function find_divisor
	#-----------------------------------------------------------
	n == smallest_divisor()::Signed
end # function myIsprime3

# ╔═╡ dcb6c753-3b16-4283-b445-86d1d8b97597
myIsprime3(1)                        # ==> true --> :)

# ╔═╡ 982407df-7197-4999-a674-fdfb17cacd4a
myIsprime3(2)                        # ==> true --> :)

# ╔═╡ 5a5c83a9-6916-4aeb-901a-42485cde0798
myIsprime3(3)                        # ==> true --> :)

# ╔═╡ 03af5d66-77d1-4a7c-bba6-6982ddef3595
myIsprime3(4)                        # ==> false --> :)

# ╔═╡ 74025279-68a4-4370-81f2-d68cf8002505
myIsprime3(5)                        # ==> true --> :)

# ╔═╡ d226bb12-6728-471b-8281-4e6d22e2c75a
myIsprime3(6)                        # ==> false --> :)

# ╔═╡ d9f2f821-8d37-40df-b810-92c4bba4d0bd
myIsprime3(97)                       # ==> true --> :)

# ╔═╡ e4bfcf03-0fed-489e-b582-d63eb339651e
myIsprime3(99989)                    # ==> true --> :)

# ╔═╡ 85683cc3-a96a-40e5-9136-7e954bca0be7
myIsprime3(99991)                    # ==> true --> :)

# ╔═╡ bbc0425b-076b-4a5a-8fce-853845c5a26e
myIsprime3(99993)                    # ==> false --> :)

# ╔═╡ e7e74706-cdf4-46d0-945d-807e0d13312c
myIsprime3(100003)                   # ==> true --> :)

# ╔═╡ 661f9a76-4a93-4cd8-85d5-7cc670027be6
md"
###### [Carmichael numbers](https://en.wikipedia.org/wiki/Carmichael_number) (= [Fermat *pseudo*primes](https://en.wikipedia.org/wiki/Fermat_pseudoprime))
A *Carmichael number* will *usually* pass *Fermat primality test* even though it is *not* actually prime. But this test $isprime3$ is searching divisors *exhausively*, so its perfect but not effective.
"

# ╔═╡ 3e1f9d8e-8e07-42b6-b163-5721e219e0d7
md"
###### 1st Carmichael number
"

# ╔═╡ 5e1b7f4e-837c-4b87-9da8-0641b0f3a71b
myIsprime3(561)                       # ==> false --> :)

# ╔═╡ 702cbddc-d362-4029-95eb-b0589ebec23e
md"
###### 2nd Carmichael number
"

# ╔═╡ 6f3e5541-0d66-43ee-9341-d88d29bdca72
myIsprime3(1105)                      # ==> false --> :)

# ╔═╡ 76c13f36-ae76-4173-b8c9-36f261236059
md"
###### 3rd Carmichael number
"

# ╔═╡ 0c88ebf7-105b-4015-a3bb-92db767c3e38
myIsprime3(1729)                      # ==> false --> :)

# ╔═╡ 4fa732fa-b031-48ad-b846-53d14ee48877
md"
###### 4th Carmichael number
"

# ╔═╡ cf3848fd-41fb-4c0d-b6b7-78907956a1e8
myIsprime3(2465)                      # ==> false --> :)

# ╔═╡ 69b961e8-1be0-4da6-9c1f-54970bcadff0
md"
###### 5th Carmichael number
"

# ╔═╡ ad93c8b8-913f-40ae-b872-e5c533e7ac91
myIsprime3(2821)                      # ==> false --> :)

# ╔═╡ d311e7c8-1136-42e3-b2b2-ed92ebd10c7a
md"
###### 6th Carmichael number
"

# ╔═╡ 1fb22733-3afa-4611-aabd-1d4ffef42d1b
myIsprime3(6601)                      # ==> false --> :)

# ╔═╡ 4be6019b-f5f8-4965-9a55-334a27b58306
md"
###### 7th Carmichael number
"

# ╔═╡ cb97dd4b-e711-4787-a5ca-5ed2f7652b29
myIsprime3(8911)                      # ==> false --> :)

# ╔═╡ af046953-fb2d-4825-b9e5-1029c0da7ac5
md"
###### 8th Carmichael number
"

# ╔═╡ 9db0c031-f53f-44cc-a1bc-6428fd86bec8
myIsprime3(10585)                     # ==> false --> :)

# ╔═╡ 6f211c92-c6e7-4b4b-88df-2ede20e06340
md"
###### 9th Carmichael number
"

# ╔═╡ e80c19fb-6868-42d7-820d-d78703659de7
myIsprime3(15841)                     # ==> false --> :)

# ╔═╡ a5782ca5-e8b0-4fd9-a710-5f54dc244cbc
md"
###### 10th Carmichael number
"

# ╔═╡ 92c1fd0b-0d46-4b42-a975-5f4a6933ec6b
myIsprime3(29341)                     # ==> false --> :)

# ╔═╡ 9af4ac3a-b262-4e4a-a9d7-9eb3f26f81db
md"
###### 11th Carmichael number
"

# ╔═╡ 3f8afe4b-f381-4979-9e7d-b1c2883a2c85
myIsprime3(41041)                     # ==> false --> :)

# ╔═╡ f78510e7-aa45-4a15-8dcd-9e27052d68ea
md"
###### 12th Carmichael number
"

# ╔═╡ a344b574-3520-414b-887a-a55ed91334a7
myIsprime3(46657)                     # ==> false --> :)

# ╔═╡ 4880be53-a679-4408-94de-d33a500bf216
md"
###### 13th Carmichael number
"

# ╔═╡ 678bf3f6-d974-4571-8cdb-ac766ded79f1
myIsprime3(52633)                     # ==> false --> :)

# ╔═╡ 59b4a0c5-2c86-41f7-8513-aa6f3a5f68c2
md"
###### 14th Carmichael number
"

# ╔═╡ 15e7f4ad-8a86-4f8d-9d4e-61fb6f747f09
myIsprime3(62745)                     # ==> false --> :)

# ╔═╡ 31d6b5b8-9bc7-4e49-8db1-c65c1b909b79
md"
###### 15th Carmichael number
"

# ╔═╡ c629e780-04cf-4198-bc1e-09052d50d630
myIsprime3(63973)                     # ==> false --> :)

# ╔═╡ 5d64c3d0-0179-457f-8154-f4e8f203316f
md"
###### 16th Carmichael number
"

# ╔═╡ 39b8633f-2fa4-4918-a98e-ccf6f5aa5bf1
myIsprime3(75361)                     # ==> false --> :)

# ╔═╡ ee5289aa-835d-4346-9294-b3c38f9dcd1c
md"
---
###### 4.2 Iterative Function $isprime4$ 
with localized definitions and function-wide scope of $n$

"

# ╔═╡ c665c4fe-4061-4da1-a324-a43c126db248
√110

# ╔═╡ 7ca56b5e-2662-44fd-9451-d13dd5b69164
Int

# ╔═╡ 99ce7b3d-0f79-4495-a786-b42fa173e992
round(Int, √2)

# ╔═╡ 45301594-f549-4897-bb87-8b78e7778746
round(Int, √3)

# ╔═╡ aa546fa7-be8b-4768-9198-0e16bde99e46
round(Int, √4)

# ╔═╡ 00b9a040-f24a-4c9a-b0ae-3896b994343d
round(Int, √110)

# ╔═╡ 4eb6cc09-f864-4a80-9d8d-a03a08e0f97e
let count = 0
	for i = 1:100
		count = i
		i > 5 && break # if ... then ...
	end # for
	count
end # let

# ╔═╡ 2beafb91-5227-45eb-bb09-138128713268
# function-wide scope of 'n';  suppressed in local functions
function myIsprime4(n::Signed)::Bool
	#------------------------------------------------------
	divides(a::Signed, b::Signed)::Bool = rem(b, a) == 0 
	#------------------------------------------------------
	function smallest_divisor()::Signed
		test_divisor = 2
		for i = 2:round(Int,√n)
			test_divisor = i
			divides(test_divisor, n) && break
		end # for
		divides(test_divisor, n) ? test_divisor : n
	end # function smallest_divisor
	#------------------------------------------------------
	n == smallest_divisor() 
end # function myIsprime4

# ╔═╡ 661fdf22-22f5-4b7f-aec4-04422ec3c50d
myIsprime4(1)                        # ==> true --> :)

# ╔═╡ 00a795d7-fe31-4568-b53a-7511052e6e06
myIsprime4(2)                        # ==> true --> :)

# ╔═╡ 83a17a23-8486-476b-9b2d-fd6203c03ff9
myIsprime4(3)                        # ==> true --> :)

# ╔═╡ 2885dc3c-e984-4008-a153-e4a438e398d8
myIsprime4(4)                        # ==> false --> :)

# ╔═╡ af776524-b603-408b-966e-6d7b2d462f73
myIsprime4(5)                        # ==> true --> :)

# ╔═╡ 3048bc35-c1ed-44fd-b761-7a0f7b65ac67
myIsprime4(6)                        # ==> false --> :)

# ╔═╡ 9ae6d8e7-3c27-4b74-8f1c-57057dc525a1
myIsprime4(97)                       # ==> true --> :)

# ╔═╡ b7aa0853-2b58-4689-b0a2-bc601cff7174
myIsprime4(99989)                    # ==> true --> :)

# ╔═╡ ef11bdd6-ac98-4f26-afa7-142dae6a214a
myIsprime4(99991)                    # ==> true --> :)

# ╔═╡ 64f76aea-c240-45d2-b146-42007a18ae89
myIsprime4(99993)                    # ==> false --> :)

# ╔═╡ 03a54358-3228-48f2-a4d6-5598704b5df6
myIsprime4(100003)                   # ==> true --> :)

# ╔═╡ 0eb278bf-4ded-41f7-ad7e-fd4fd4de8342
md"
###### [Carmichael numbers](https://en.wikipedia.org/wiki/Carmichael_number) (= [Fermat *pseudo*primes](https://en.wikipedia.org/wiki/Fermat_pseudoprime))
A *Carmichael number* will *usually* pass *Fermat primality test* even though it is *not* actually prime. But this test $myIsprime4$ is searching divisors *exhausively*, so its perfect but not effective.
"

# ╔═╡ 59d7d87b-850d-4934-84ed-985ebf068794
md"
###### 1st Carmichael number
"

# ╔═╡ cea4063a-2f10-4cb7-944b-a2055d41c3d1
myIsprime4(561)                       # ==> false --> :)

# ╔═╡ 77b86d25-3640-4455-a66f-c9f6fff4f75b
md"
###### 2nd Carmichael number
"

# ╔═╡ 50e1140a-f108-4236-8b43-989ab458ef95
myIsprime4(1105)                      # ==> false --> :)

# ╔═╡ 4c174e06-3812-42b4-bdb5-4cc771ad10f5
md"
###### 3rd Carmichael number
"

# ╔═╡ b6a6ffa9-31be-4196-9b6d-680c1406ef4a
myIsprime4(1729)                      # ==> false --> :)

# ╔═╡ 1cb7e0dd-d7e8-4f79-b3ae-5f7bbde93f35
md"
###### 4th Carmichael number
"

# ╔═╡ 509ffec5-f06a-4fd9-9723-9c5116104df7
myIsprime4(2465)                      # ==> false --> :)

# ╔═╡ d3d45287-4949-4a89-ab93-43109e3e8bb6
md"
###### 5th Carmichael number
"

# ╔═╡ aee21fdd-e1ca-49bd-b208-c1bcbaf258b6
myIsprime4(2821)                      # ==> false --> :)

# ╔═╡ 6b67cf55-514d-492a-aa89-50eda8a19c82
md"
###### 6th Carmichael number
"

# ╔═╡ a0e3b5a1-611a-4b8f-80ee-15d5ab3725ef
myIsprime4(6601)                      # ==> false --> :)

# ╔═╡ 6802ad17-0a23-42c6-876a-6ed750743acd
md"
###### 7th Carmichael number
"

# ╔═╡ 41215219-cd82-4b7d-85b6-4a985f34aa18
myIsprime4(8911)                      # ==> false --> :)

# ╔═╡ e3c37e74-1bff-4d8c-8bb2-a27d918254ea
md"
###### 8th Carmichael number
"

# ╔═╡ 886d9958-8d7a-40ca-bde0-bccc20d54c4b
myIsprime4(10585)                     # ==> false --> :)

# ╔═╡ 66c9d14d-7870-4c1c-9a1f-94a3cf23bfe0
md"
###### 9th Carmichael number
"

# ╔═╡ 2726b89a-325d-4675-b896-922392a5fef9
myIsprime4(15841)                     # ==> false --> :)

# ╔═╡ 992111fa-b073-421c-9799-a83b0d96fda1
md"
###### 10th Carmichael number
"

# ╔═╡ 2cfc5a17-7e12-4571-ab1a-c37219fe50d9
myIsprime4(29341)                     # ==> false --> :)

# ╔═╡ 9c29a2e7-e877-44af-926c-7c2c0df6023d
md"
###### 11th Carmichael number
"

# ╔═╡ 7b7b6ccf-5635-4c7a-93c3-12f1ab7519dd
myIsprime4(41041)                     # ==> false --> :)

# ╔═╡ f39e0478-8094-4049-9eb0-3bafe652ad78
md"
###### 12th Carmichael number
"

# ╔═╡ 81c2762b-4253-4929-a603-cf5ce8af292f
myIsprime4(46657)                     # ==> false --> :)

# ╔═╡ 2ebd4469-4492-4cc6-97ef-9498069fd66a
md"
###### 13th Carmichael number
"

# ╔═╡ 339c5cf9-2bd6-478c-aa79-2ab9323376b4
myIsprime4(52633)                     # ==> false --> :)

# ╔═╡ 6641f813-6e0f-41c6-a4ff-2bd402369536
md"
###### 14th Carmichael number
"

# ╔═╡ 9218057c-7372-452b-914a-641330012c43
myIsprime4(62745)                     # ==> false --> :)

# ╔═╡ 596887c2-5e6d-481f-ac70-4141e1d70c35
md"
###### 15th Carmichael number
"

# ╔═╡ 02ae9603-dc76-4c4f-a403-3d3355c3782d
myIsprime4(63973)                     # ==> false --> :)

# ╔═╡ e4b5b0a1-6f99-4af6-9c1d-43f39b211886
md"
###### 16th Carmichael number
"

# ╔═╡ db54ef5a-4c41-42c5-b89a-71edd5a861af
myIsprime4(75361)                     # ==> false --> :)

# ╔═╡ b06561f2-546b-44a7-a4a3-43056d1019e4
md"
---
###### 4.3 Iterative *Fermat Test*: $fast\_prime3$
"

# ╔═╡ b1601460-c2d3-492a-85cc-c3cd08a37546
function fast_prime3(n::Integer, times::Integer)::Bool
	#--------------------------------------------
	# while !(times == 0 || !fermat_test2(n))  
	#  ≡ while times !== 0 && fermat_test2(n)
	#--------------------------------------------
	while (times != 0) && fermat_test2(n)
		times -= 1
	end
	times == 0 ? true : false
end # function fast_prime3

# ╔═╡ 136294ce-8c08-4607-ab0b-c1fd1fdb9e44
fast_prime3(2, 100)           # ==> true --> :)

# ╔═╡ 1f1a7137-3856-4392-a731-06e2a5dab683
fast_prime3(3, 100)           # ==> true --> :)

# ╔═╡ c3757944-71d2-4783-8dad-12093b0ddfc6
fast_prime3(4, 100)           # ==> false --> :)

# ╔═╡ dbda9f3f-1a59-4fd1-9672-1fefa020efb5
fast_prime3(5, 100)           # ==> true --> :)

# ╔═╡ e485954f-ec8d-46db-b0f6-e8ff7c545e60
fast_prime3(6, 100)           # ==> false --> :)

# ╔═╡ 049602a5-b86a-4975-a038-1429451e154e
fast_prime3(7, 100)           # ==> true --> :)

# ╔═╡ b3da196b-748c-4018-8aea-7f25cea226d4
fast_prime3(9, 100)           # ==> false --> :)

# ╔═╡ d6fe0d38-1d59-41b8-8895-df28eb1cf49a
fast_prime3(11, 100)          # ==> true --> :)

# ╔═╡ 67683aa1-bb33-4f39-b6ce-4f0c17ca94e4
fast_prime3(97, 100)          # ==> true --> :)

# ╔═╡ 25a17f89-0568-4f40-93ee-c66b49824ea7
fast_prime3(99989, 100)       # ==> true --> :)

# ╔═╡ dc86cd34-db60-4472-917d-164dfb017154
fast_prime3(99991, 100)       # ==> true --> :)

# ╔═╡ 2f4a0498-bc9f-48bf-adaa-36d096fa3bf1
fast_prime3(99993, 100)       # ==> false --> :), because 3*33331 = 99993

# ╔═╡ 488b4b80-7bab-4bf0-9d91-6bc7d638476a
fast_prime3(100003, 100)      # ==> true --> :)

# ╔═╡ ea8bafa9-ab4b-477e-b702-76cfa1770e5c
md"
###### Carmichael numbers *usually* fool Fermat's Test, but not $fast\_prime3$
"

# ╔═╡ aa470a5e-d5dd-4575-9380-a6fe948fdafd
md"
###### 1st Carmichael number
"

# ╔═╡ 435fc3f4-022f-4d76-bfce-a059d422ea19
fast_prime2(561, 100)        # ==> false --> :) ; test could *not* be fooled

# ╔═╡ e53609bd-0200-41cf-884d-c3b55790f6a2
md"
###### 2nd Carmichael number
"

# ╔═╡ e84189b9-0f30-44b0-a5bf-db9e4a95ca3d
fast_prime2(1105, 100)       # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 6c17f1e3-c604-4eb7-9c8e-c3e599aa658b
md"
###### 3rd Carmichael number
"

# ╔═╡ dc1fb0d7-099f-4f97-aded-39fa887bdfef
fast_prime2(1729, 100)       # ==> false --> :) ; test could *not* be fooled

# ╔═╡ ed89e6f0-6beb-4a56-b5d8-234ed99e1705
md"
###### 4th Carmichael number
"

# ╔═╡ 16646d37-f82c-455a-9baf-78e262b863e2
fast_prime2(2465, 100)       # ==> false --> :) ; test could *not* be fooled

# ╔═╡ d2db7b42-7740-4cfc-a7ab-774e1cd9a295
md"
###### 5th Carmichael number
"

# ╔═╡ 828970a9-1284-4b3f-9d83-9768d959fde9
fast_prime2(2821, 100)       # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 3f0b976a-2c05-4759-b35e-8eb23fd3b59c
md"
###### 6th Carmichael number
"

# ╔═╡ b6d25de8-9f4d-41b6-96f7-684e2d583ade
fast_prime2(6601, 100)       # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 0a99ecf0-40ff-4859-94d5-daa23a8ff1ad
md"
###### 7th Carmichael number
"

# ╔═╡ f3a80c96-41df-4fa0-8ded-3d247d16cf4c
fast_prime2(8911, 100)       # ==> false --> :) ; test could *not* be fooled

# ╔═╡ a12b1bf2-8ce1-4533-b6ba-f9330225a92e
md"
###### 8th Carmichael number
"

# ╔═╡ b88083e3-8a6b-46a7-b024-38f84756b987
fast_prime2(10585, 100)      # ==> false --> :) ; test could *not* be fooled

# ╔═╡ b21ea350-86fd-4225-a8b7-5ba27dd38e8f
md"
###### 9th Carmichael number
"

# ╔═╡ 763a0b2a-934c-4cae-bddf-001d2f2cebee
fast_prime2(15841, 100)      # ==> false --> :) ; test could *not* be fooled

# ╔═╡ ee88876a-a04e-4bba-bf15-fab521747859
md"
###### 10th Carmichael number
"

# ╔═╡ d9805d69-d9f6-465c-a795-593f6b6a1faa
fast_prime2(29341, 100)      # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 71c32ec8-9a8a-40f9-9bdb-39c5bc1a1f0e
md"
###### 11th Carmichael number
"

# ╔═╡ 281f123a-8455-4f69-919e-2e61f0985546
fast_prime2(41041, 100)      # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 062175aa-0057-4518-9560-d2847b3a2627
md"
###### 12th Carmichael number
"

# ╔═╡ e8ecccf2-224a-43d7-b8bd-98011d00def9
fast_prime2(46657, 100)      # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 670c7bb1-0ba7-4aa4-ac3f-b6f4dee02cd0
md"
###### 13th Carmichael number
"

# ╔═╡ e8f244c1-09e0-4782-917f-d8435e6ed1fa
fast_prime2(52633, 100)      # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 7ee9d52a-4b8a-4baf-86be-4b9c4d6f98ae
md"
###### 14th Carmichael number
"

# ╔═╡ 89d56dbd-9275-4137-87dc-1731272ed88b
fast_prime2(62745, 100)      # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 0cf91598-9774-41b8-90aa-e53da69c464f
md"
###### 15th Carmichael number
"

# ╔═╡ ca30da6b-5965-44c8-88c8-580e6baf92cb
fast_prime2(63973, 100)      # ==> false --> :) ; test could *not* be fooled

# ╔═╡ fd5a057b-5f8f-4d4b-8d8f-68794ad2d46f
md"
###### 16th Carmichael number
"

# ╔═╡ 0f9ee0d8-2f73-43ee-bc67-279342f96168
fast_prime2(75361, 100)      # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 27648379-3689-414c-a126-2e00921beb78
md"
---
##### 5. Julia Package function $Primes.isprime$
"

# ╔═╡ afb70559-ed53-49b5-87ef-d0c9ec45ff60
Primes.isprime(1)               # ==> false --> :)

# ╔═╡ 1fffc29f-dcf1-4e25-9922-30c19e64d208
Primes.isprime(2)               # ==> true --> :); 2 is prime

# ╔═╡ a2efacd3-76a4-4afb-b33d-86231076fc5d
Primes.isprime(3)               # ==> true --> :); 3 is prime

# ╔═╡ f7b74f24-f84a-4a20-9e24-ac5121e8ea09
Primes.isprime(4)               # ==> false --> :)

# ╔═╡ 2b460f2f-2315-49e2-886c-eb2eaf3e3927
Primes.isprime(5)               # ==> true --> :); 5 is prime

# ╔═╡ 64a34c97-df81-4d75-8b66-ef611d7bfe13
Primes.isprime(6)               # ==> false --> :)

# ╔═╡ 38bf06b5-f147-4654-90c4-4c441be4aa72
Primes.isprime(97)              # ==> true --> :); 97 is prime

# ╔═╡ 58b5599f-b6a2-4d02-8b0f-2a7b11145986
md"
###### Sample from the [First 100,008 Primes](https://primes.utm.edu/lists/small/100000.txt)
"

# ╔═╡ 253ccb6c-2b96-44c0-9691-8a943314c53d
Primes.isprime(99989)           # ==> true --> :)

# ╔═╡ d38ce7d0-0c93-4513-882f-1a74230f1063
Primes.isprime(99991)           # ==> true --> :)

# ╔═╡ eeead12a-70d8-4ebf-b41d-60694a031010
Primes.isprime(99993)           # ==> false --> :); because 3 * 33331 = 99993

# ╔═╡ 438e35e7-91d8-412a-8514-c6c5b4b269c6
Primes.isprime(100003)          # ==> true --> :)

# ╔═╡ 2339ae20-64d9-46a6-bbb6-71a5587cf558
md"
###### Carmichael numbers *usually* fool Fermat's Test, but not $Primes.isprime$
"

# ╔═╡ e0d4084c-436a-4def-8e65-a944dc522f8b
md"
###### 1st Carmichael number
"

# ╔═╡ 8fc835d4-bec7-4026-b17c-b23aac89aae3
Primes.isprime(561)             # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 639e2cfc-2fb6-4e43-85bc-ef55ae167949
md"
###### 2nd Carmichael number
"

# ╔═╡ 40860255-4a68-4f0b-a8e2-7dabefb6c497
Primes.isprime(1105)            # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 05ec46f4-128c-4628-90d7-bf5818173f58
md"
###### 3rd Carmichael number
"

# ╔═╡ 8cb9af71-dcea-4b83-a5ab-8713b6c78ca2
Primes.isprime(1729)            # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 160fd108-a0a7-4d61-a11e-c0ceb2be16e6
md"
###### 4th Carmichael number
"

# ╔═╡ 20a1747b-0949-4d5e-91a8-34f9c24562a0
Primes.isprime(2465)            # ==> false --> :) ; test could *not* be fooled

# ╔═╡ a184f35c-b5b2-44b7-bbbd-eeedebe14fa5
md"
###### 5th Carmichael number
"

# ╔═╡ 5ed16f5b-9550-4149-bf78-c1faa5e3e6e9
Primes.isprime(2821)            # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 093d2b2e-5be7-476b-b16f-a5f0edf92b29
md"
###### 6th Carmichael number
"

# ╔═╡ 3952f23b-366c-46b4-889b-a903c913a236
Primes.isprime(6601)            # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 2f64673d-798c-4ca1-b515-4bcb537b5316
md"
###### 7th Carmichael number
"

# ╔═╡ 000b6a03-6d41-40d2-9d4a-ef84d09ffe41
Primes.isprime(8911)            # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 3272a62e-cd81-4be2-a4cc-ec0ca276344b
md"
###### 8th Carmichael number
"

# ╔═╡ f9cc80f9-df9f-4c05-88a1-1d80d5b039b9
Primes.isprime(10585)           # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 7e11f80d-aa82-4525-a43e-bc09b663db68
md"
###### 9th Carmichael number
"

# ╔═╡ 3841b3de-11ef-4912-b886-3a24a401d429
Primes.isprime(15841)           # ==> false --> :) ; test could *not* be fooled

# ╔═╡ a52fd122-e1a0-4b50-bb23-ae8ecc593118
md"
###### 10th Carmichael number
"

# ╔═╡ a8aea5f3-336e-41fd-b670-679e3f1d9a65
Primes.isprime(29341)           # ==> false --> :) ; test could *not* be fooled

# ╔═╡ d388dec0-51e3-4d4c-a487-fde7684f7059
md"
###### 11th Carmichael number
"

# ╔═╡ a20afae0-d9ec-4e16-9118-c939b92665b1
Primes.isprime(41041)           # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 9d2e01a5-2a6d-43bf-a2c7-2fd85512b103
md"
###### 12th Carmichael number
"

# ╔═╡ b8e45861-2935-4253-8178-e6a978ba8ed1
Primes.isprime(46657)           # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 9479588a-f884-4a3f-9b4d-c33509eb67b8
md"
###### 13th Carmichael number
"

# ╔═╡ 04e589f3-8703-4f17-b4b5-764abe7be9ec
Primes.isprime(52633)           # ==> false --> :) ; test could *not* be fooled

# ╔═╡ e22eb7f1-c7df-4a41-9a9b-92b114994137
md"
###### 14th Carmichael number
"

# ╔═╡ 17b4ff5a-c78d-4f66-9db7-03c792d0e094
Primes.isprime(62745)           # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 80c66cb1-3f37-4561-8e70-a6a0129d0f29
md"
###### 15th Carmichael number
"

# ╔═╡ 9dceb743-767a-4fac-aeb1-83e601e989be
Primes.isprime(63973)           # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 4de5d117-4720-4319-8283-ba4e5a808604
md"
###### 16th Carmichael number
"

# ╔═╡ eb91249c-2a72-4643-9d80-e87d80f0cf58
Primes.isprime(75361)           # ==> false --> :) ; test could *not* be fooled

# ╔═╡ 6626ffdb-fdc0-492b-b7e8-dcf4a98afae7
md"
---
##### 6. Summary
We presented four methods for identifying primes. The first is the *exhaustive search for divisors*. The next two are based on *Fermat's Little Theorem*. The first is based on the *longer* form of the theorem and a transpilation of *SICP*'s Scheme function. The second is based on a *shorter* version of *Fermat's Little Theorem*. The last is a Julia package function $Primes.isprime$.

The *first* algorithm is perfect but inefficient. The *second* can be fooled by all 16 Carmichel numbers we studied. The *third* and the *fourth* are perfect in this respect. All 16 Carmichel numbers could be unmasked as *composites*.

"

# ╔═╡ 5889eede-0150-476a-a0d1-7298b616172f
md"
---
##### 7. References
- **Abelson, H., Sussman, G.J. & Sussman, J.**; [*Structure and Interpretation of Computer Programs*](https://mitp-content-server.mit.edu/books/content/sectbyfn/books_pres_0/6515/sicp.zip/full-text/book/book.html), Cambridge, Mass.: MIT Press, (2/e), 1996; last visit 2025/01/21

- **Abelson, H., Sussman, G.J. & Sussman, J.**; [*Structure and Interpretation of Computer Programs*](https://web.mit.edu/6.001/6.037/sicp.pdf); Cambridge, Mass.: MIT Press, (2/e), 2016; last visit 2025/01/21

- **Cormen, Th.H., Leiserson, Ch.e., Rivest, R.L. & Stein, C.**; *Introduction to Algorithms*, Cambridge, Mass.: MIT Press, 2022, (4/e)

- **t5k.org**; [*List of 100.008 Primes*](https://t5k.org/lists/small/100000.txt), last visit 2025/01/26

- **Wikipedia**; [*Carmichael Number*](https://en.wikipedia.org/wiki/Carmichael_number); last visit 2025/01/26

- **Wikipedia**; [*Euler's totient function*](https://en.wikipedia.org/wiki/Euler%27s_totient_function): last visit 2025/02/01

- **Wikipedia**; [*Fermat's little theorem*](https://en.wikipedia.org/wiki/Fermat%27s_little_theorem); last visit 2025/01/27

- **Wikipedia**; [*Likelihood*](https://en.wikipedia.org/wiki/Likelihood_function); last visit 2025/01/28

- **Wikipedia**; [*Prime Number*](https://en.wikipedia.org/wiki/Prime_number); last vist 2025/02/02

- **Wikipedia**; [*Pseudoprimes*](https://en.wikipedia.org/wiki/Pseudoprime); last visit 2025/01/28

- **Wikipedia**; [*The First 100,008 Primes*](https://en.wikipedia.org/wiki/List_of_prime_numbers); last visit 2025/01/21

"

# ╔═╡ 5ddec9aa-8c2c-447f-b97c-a91b39be792b
md"
----
###### end of ch. 1.2.6
==================================================================================

This is a **draft** under the [Attribution-NonCommercial-ShareAlike 4.0 International **(CC BY-NC-SA 4.0)**](https://creativecommons.org/licenses/by-nc-sa/4.0/deed.en) license; Comments, suggestions for improvement and bug reports are welcome: **claus.moebus(@)uol.de**

==================================================================================
"

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    ConstructionBase = "187b0558-2788-49d3-abe0-74a17ed4e7c9"
    InverseFunctions = "3587e190-3f89-42d0-90ee-14403ec27112"

[[deps.UnitfulLatexify]]
deps = ["LaTeXStrings", "Latexify", "Unitful"]
git-tree-sha1 = "975c354fcd5f7e1ddcc1f1a23e6e091d99e99bc8"
uuid = "45397f5d-5981-4c77-b2b3-fc36d6e9b728"
version = "1.6.4"

[[deps.Unzip]]
git-tree-sha1 = "ca0969166a028236229f63514992fc073799bb78"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.2.0"

[[deps.Vulkan_Loader_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Wayland_jll", "Xorg_libX11_jll", "Xorg_libXrandr_jll", "xkbcommon_jll"]
git-tree-sha1 = "2f0486047a07670caad3a81a075d2e518acc5c59"
uuid = "a44049a8-05dd-5a78-86c9-5fde0876e88c"
version = "1.3.243+0"

[[deps.Wayland_jll]]
deps = ["Artifacts", "EpollShim_jll", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "85c7811eddec9e7f22615371c3cc81a504c508ee"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.21.0+2"

[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "5db3e9d307d32baba7067b13fc7b5aa6edd4a19a"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.36.0+0"

[[deps.WoodburyMatrices]]
deps = ["LinearAlgebra", "SparseArrays"]
git-tree-sha1 = "c1a7aa6219628fcd757dede0ca95e245c5cd9511"
uuid = "efce3f68-66dc-5838-9240-27a6d6f5f9b6"
version = "1.0.0"

[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Zlib_jll"]
git-tree-sha1 = "a2fccc6559132927d4c5dc183e3e01048c6dcbd6"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.13.5+0"

[[deps.XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "7d1671acbe47ac88e981868a078bd6b4e27c5191"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.42+0"

[[deps.XZ_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "15e637a697345f6743674f1322beefbc5dcd5cfc"
uuid = "ffd25f8a-64ca-5728-b0f7-c24cf3aae800"
version = "5.6.3+2"

[[deps.Xorg_libICE_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "326b4fea307b0b39892b3e85fa451692eda8d46c"
uuid = "f67eecfb-183a-506d-b269-f58e52b52d7c"
version = "1.1.1+0"

[[deps.Xorg_libSM_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libICE_jll"]
git-tree-sha1 = "3796722887072218eabafb494a13c963209754ce"
uuid = "c834827a-8449-5923-a945-d239c165b7dd"
version = "1.2.4+0"

[[deps.Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "9dafcee1d24c4f024e7edc92603cedba72118283"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.8.6+3"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "2b0e27d52ec9d8d483e2ca0b72b3cb1a8df5c27a"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.11+3"

[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "807c226eaf3651e7b2c468f687ac788291f9a89b"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.3+0"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "02054ee01980c90297412e4c809c8694d7323af3"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.4+3"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "d7155fea91a4123ef59f42c4afb5ab3b4ca95058"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.6+3"

[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "6fcc21d5aea1a0b7cce6cab3e62246abd1949b86"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "6.0.0+0"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "984b313b049c89739075b8e2a94407076de17449"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.8.2+0"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXext_jll"]
git-tree-sha1 = "a1a7eaf6c3b5b05cb903e35e8372049b107ac729"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.5+0"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "b6f664b7b2f6a39689d822a6300b14df4668f0f4"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.4+0"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "a490c6212a0e90d2d55111ac956f7c4fa9c277a6"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.11+1"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "fee57a273563e273f0f53275101cd41a8153517a"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.1+3"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "1a74296303b6524a0472a8cb12d3d87a78eb3612"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.17.0+3"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "dbc53e4cf7701c6c7047c51e17d6e64df55dca94"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.2+1"

[[deps.Xorg_xcb_util_cursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_jll", "Xorg_xcb_util_renderutil_jll"]
git-tree-sha1 = "04341cb870f29dcd5e39055f895c39d016e18ccd"
uuid = "e920d4aa-a673-5f3a-b3d7-f755a4d47c43"
version = "0.1.4+0"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "ab2221d309eda71020cdda67a973aa582aa85d69"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.6+1"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "691634e5453ad362044e2ad653e79f3ee3bb98c3"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.39.0+0"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "b9ead2d2bdb27330545eb14234a2e300da61232e"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.5.0+3"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+1"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "622cf78670d067c738667aaa96c553430b65e269"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.7+0"

[[deps.eudev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "gperf_jll"]
git-tree-sha1 = "431b678a28ebb559d224c0b6b6d01afce87c51ba"
uuid = "35ca27e7-8b34-5b7f-bca9-bdc33f59eb06"
version = "3.2.9+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "6e50f145003024df4f5cb96c7fce79466741d601"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.56.3+0"

[[deps.gperf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "0ba42241cb6809f1a278d0bcb976e0483c3f1f2d"
uuid = "1a1c6b14-54f6-533d-8383-74cd7377aa70"
version = "3.1.1+1"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "522c1df09d05a71785765d19c9524661234738e9"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.11.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "e17c115d55c5fbb7e52ebedb427a0dca79d4484e"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.2+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.11.0+0"

[[deps.libdecor_jll]]
deps = ["Artifacts", "Dbus_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pango_jll", "Wayland_jll", "xkbcommon_jll"]
git-tree-sha1 = "9bf7903af251d2050b467f76bdbe57ce541f7f4f"
uuid = "1183f4f0-6f2a-5f1a-908b-139f9cdfea6f"
version = "0.2.2+0"

[[deps.libevdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "141fe65dc3efabb0b1d5ba74e91f6ad26f84cc22"
uuid = "2db6ffa8-e38f-5e21-84af-90c45d0032cc"
version = "1.11.0+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "8a22cf860a7d27e4f3498a0fe0811a7957badb38"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.3+0"

[[deps.libinput_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "eudev_jll", "libevdev_jll", "mtdev_jll"]
git-tree-sha1 = "ad50e5b90f222cfe78aa3d5183a20a12de1322ce"
uuid = "36db933b-70db-51c0-b978-0f229ee0e533"
version = "1.18.0+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "b7bfd3ab9d2c58c3829684142f5804e4c6499abc"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.45+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "490376214c4721cdaca654041f635213c6165cb3"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+2"

[[deps.mtdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "814e154bdb7be91d78b6802843f76b6ece642f11"
uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e"
version = "1.1.6+0"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.59.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+2"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "63406453ed9b33a0df95d570816d5366c92b7809"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+2"
"""

# ╔═╡ Cell order:
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