solve 41 and 46

Src/Problem40.lean

@@ -3,17 +3,24 @@
 (Intermediate 🌟🌟) Goldbach's conjecture says that every positive even number greater than `2` is the sum of two prime numbers. Example: `28 = 5 + 23`. It is one of the most famous facts in number theory that has not been proved to be correct in the general case. It has been numerically confirmed up to very large numbers (much larger than we can go with our Prolog system). Write a predicate to find the two prime numbers that sum up to a given even integer.
 -/
 
-def Nat.isPrime (n : Nat) : Bool :=
-  if n == 0 || n == 1 then
-    false
-  else
-    let properDivisors := List.range n
-      |>.drop 2
-      |>.filter (n % · == 0)
-    properDivisors.length == 0
+def Nat.isPrime (n : Nat) : Bool := Id.run do
+  if n ≤ 2 then
+    return false
+  for d in [2:n] do
+    if n % d = 0 then
+      return false
+    if d ^ 2 > n then
+      break
+  return true
+
+-- You can use this!
+#check Nat.isPrime
 
 def goldbach (n : Nat) : Nat × Nat := Id.run do
   -- sorry
+  if n % 2 ≠ 0 then
+    panic! "n must be an even number"
+
   for cand in (List.range n) do
     if not cand.isPrime then
       continue
@@ -41,3 +48,5 @@ def goldbachTest (n : Nat) : IO Unit :=
 #eval goldbachTest 308
 
 #eval goldbachTest 308000
+
+#eval goldbachTest 19278020

Src/Problem41.lean

@@ -0,0 +1,42 @@
+/- # Problem 40
+
+(Intermediate 🌟🌟) A list of even numbers and their Goldbach compositions in a given range.
+-/
+
+def Nat.isPrime (n : Nat) : Bool := Id.run do
+  if n ≤ 2 then
+    return false
+  for d in [2:n] do
+    if n % d = 0 then
+      return false
+    if d ^ 2 > n then
+      break
+  return true
+
+def goldbach (n : Nat) : Nat × Nat := Id.run do
+  for cand in [2:n] do
+    if not cand.isPrime then
+      continue
+    let rest := n - cand
+    if not rest.isPrime then
+      continue
+    return (cand, rest)
+
+  panic! "we've found a counter example of goldbach conjecture!! 🎉"
+
+def goldbachList (lower upper : Nat) (primeLower : Nat := 2) : List (Nat × Nat) :=
+  -- sorry
+  List.range (upper + 1)
+    |>.filter (· ≥ lower)
+    |>.filter (· % 2 == 0)
+    |>.map (fun n => (goldbach n))
+    |>.filter (fun t => t.fst > primeLower)
+  -- sorry
+
+-- The following codes are for test and you should not edit these.
+
+#guard goldbachList 9 20 == [(3, 7), (5, 7), (3, 11), (3, 13), (5, 13), (3, 17)]
+
+#guard goldbachList 9 20 3 == [(5, 7), (5, 13)]
+
+#guard goldbachList 4 2000 50 == [(73,919),(61,1321),(67,1789),(61,1867)]

Src/Problem46.lean

@@ -0,0 +1,24 @@
+/- # Problem 46
+(Intermediate 🌟🌟) Truth tables for logical expressions.
+-/
+
+
+def table (p : Bool → Bool → Bool) : List (List Bool) :=
+  -- sorry
+  [
+    [true, true, p true true],
+    [true, false, p true false],
+    [false, true, p false true],
+    [false, false, p false false]
+  ]
+  -- sorry
+
+-- The following codes are for test and you should not edit these.
+
+#guard table (fun a b => And a (Or a b)) ==
+  [
+    [true, true, true],
+    [true, false, true],
+    [false, true, false],
+    [false, false, false]
+  ]

md/SUMMARY.md

@@ -51,4 +51,9 @@
 * [37: Euler's totient improved](./build/Problem37.md)
 * [38: Compare totient calculations]()
 * [39: Prime numbers in a range](./build/Problem39.md)
-* [40: Goldbach's conjecture](./build/Problem40.md)
+* [40: Goldbach's conjecture](./build/Problem40.md)
+* [41: Goldbach's list](./build/Problem41.md)
+
+# 46-50: Logic and codes
+
+* [46: Truth table](./build/Problem46.md)