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Idoneal numbers completeness conjecture

An integer $D>0$ is idoneal if every integer that can be expressed in exactly one way (up to order and signs) as $x^2 + D y^2$ with gcd(x, Dy)=1 is a prime power or twice a prime power.

The Idoneal Numbers Completeness Conjecture asserts that the following list of 65 numbers is complete: 1,2,3,4,5,6,7,8,9,10,12,13,15,16,18,21,22,24,25,28,30,33,37,40,42,45,48, 57,58,60,70,72,78,85,88,93,102,105,112,120,130,133,165,168,177,190,210,232, 240,253,273,280,312,330,345,357,385,408,462,520,760,840,1320,1365,1848. References:

namespace Idoneal

Equivalent definition: A positive integer $n$ is idoneal if and only if it cannot be written as $ab + bc + ac$ for distinct positive integers $a, b,$ and $c$.

def IsIdoneal (n : ℕ) : Prop := 0 < n ∧ ¬ ∃ a b c : ℕ, 0 < a ∧ a < b ∧ b < c ∧ n = a * b + b * c + a * c

The 65 known idoneal numbers that are conjectured to be the only idoneal numbers.

def knownIdonealNumbers : Finset ℕ := {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 102, 105, 112, 120, 130, 133, 165, 168, 177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, 1848}

Reduces the unbounded search for a representation n = a*b + b*c + a*c (with 0 < a < b < c) to a bounded, decidable double search over a, b ∈ range (n+1).

The third variable is not searched: for a fixed pair a, b the equation n = a*b + c*(a+b) pins down c = (n - a*b) / (a+b), so the witness c is recovered by exact division. The forward direction uses a, b ≤ n (each pairwise product is at most n) to land the pair in range (n+1).

n:ℕa:ℕb:ℕc:ℕha:0 < ahab:a < bhbc:b < cheq:n = a * b + b * c + a * chbn:b ≤ nhan:a ≤ nhsum:n - a * b = c * (a + b)hpos:0 < a + bhc:(n - a * b) / (a + b) = c⊢ ∃ a ∈ Finset.range (n + 1), ∃ b ∈ Finset.range (n + 1), 0 < a ∧ a < b ∧ b < (n - a * b) / (a + b) ∧ n = a * b + b * ((n - a * b) / (a + b)) + a * ((n - a * b) / (a + b)) exact ⟨a, Finset.mem_range.mpr (n:ℕa:ℕb:ℕc:ℕha:0 < ahab:a < bhbc:b < cheq:n = a * b + b * c + a * chbn:b ≤ nhan:a ≤ nhsum:n - a * b = c * (a + b)hpos:0 < a + bhc:(n - a * b) / (a + b) = c⊢ a < n + 1 All goals completed! 🐙), b, Finset.mem_range.mpr (n:ℕa:ℕb:ℕc:ℕha:0 < ahab:a < bhbc:b < cheq:n = a * b + b * c + a * chbn:b ≤ nhan:a ≤ nhsum:n - a * b = c * (a + b)hpos:0 < a + bhc:(n - a * b) / (a + b) = c⊢ b < n + 1 All goals completed! 🐙), ha, hab, hc ▸ hbc, hc ▸ heq⟩ n:ℕ⊢ (∃ a ∈ Finset.range (n + 1), ∃ b ∈ Finset.range (n + 1), 0 < a ∧ a < b ∧ b < (n - a * b) / (a + b) ∧ n = a * b + b * ((n - a * b) / (a + b)) + a * ((n - a * b) / (a + b))) → ∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ n = a * b + b * c + a * c n:ℕa:ℕleft✝¹:a ∈ Finset.range (n + 1)b:ℕleft✝:b ∈ Finset.range (n + 1)ha:0 < ahab:a < bhbc:b < (n - a * b) / (a + b)heq:n = a * b + b * ((n - a * b) / (a + b)) + a * ((n - a * b) / (a + b))⊢ ∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ n = a * b + b * c + a * c All goals completed! 🐙set_option maxRecDepth 4096 in

All 65 known idoneal numbers are indeed idoneal.

⊢ ¬∃ a ∈ Finset.range (1848 + 1), ∃ b ∈ Finset.range (1848 + 1), 0 < a ∧ a < b ∧ b < (1848 - a * b) / (a + b) ∧ 1848 = a * b + b * ((1848 - a * b) / (a + b)) + a * ((1848 - a * b) / (a + b)) ⊢ ¬∃ a ∈ Finset.range (1 + 1), ∃ b ∈ Finset.range (1 + 1), 0 < a ∧ a < b ∧ b < (1 - a * b) / (a + b) ∧ 1 = a * b + b * ((1 - a * b) / (a + b)) + a * ((1 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (2 + 1), ∃ b ∈ Finset.range (2 + 1), 0 < a ∧ a < b ∧ b < (2 - a * b) / (a + b) ∧ 2 = a * b + b * ((2 - a * b) / (a + b)) + a * ((2 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (3 + 1), ∃ b ∈ Finset.range (3 + 1), 0 < a ∧ a < b ∧ b < (3 - a * b) / (a + b) ∧ 3 = a * b + b * ((3 - a * b) / (a + b)) + a * ((3 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (4 + 1), ∃ b ∈ Finset.range (4 + 1), 0 < a ∧ a < b ∧ b < (4 - a * b) / (a + b) ∧ 4 = a * b + b * ((4 - a * b) / (a + b)) + a * ((4 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (5 + 1), ∃ b ∈ Finset.range (5 + 1), 0 < a ∧ a < b ∧ b < (5 - a * b) / (a + b) ∧ 5 = a * b + b * ((5 - a * b) / (a + b)) + a * ((5 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (6 + 1), ∃ b ∈ Finset.range (6 + 1), 0 < a ∧ a < b ∧ b < (6 - a * b) / (a + b) ∧ 6 = a * b + b * ((6 - a * b) / (a + b)) + a * ((6 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (7 + 1), ∃ b ∈ Finset.range (7 + 1), 0 < a ∧ a < b ∧ b < (7 - a * b) / (a + b) ∧ 7 = a * b + b * ((7 - a * b) / (a + b)) + a * ((7 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (8 + 1), ∃ b ∈ Finset.range (8 + 1), 0 < a ∧ a < b ∧ b < (8 - a * b) / (a + b) ∧ 8 = a * b + b * ((8 - a * b) / (a + b)) + a * ((8 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (9 + 1), ∃ b ∈ Finset.range (9 + 1), 0 < a ∧ a < b ∧ b < (9 - a * b) / (a + b) ∧ 9 = a * b + b * ((9 - a * b) / (a + b)) + a * ((9 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (10 + 1), ∃ b ∈ Finset.range (10 + 1), 0 < a ∧ a < b ∧ b < (10 - a * b) / (a + b) ∧ 10 = a * b + b * ((10 - a * b) / (a + b)) + a * ((10 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (12 + 1), ∃ b ∈ Finset.range (12 + 1), 0 < a ∧ a < b ∧ b < (12 - a * b) / (a + b) ∧ 12 = a * b + b * ((12 - a * b) / (a + b)) + a * ((12 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (13 + 1), ∃ b ∈ Finset.range (13 + 1), 0 < a ∧ a < b ∧ b < (13 - a * b) / (a + b) ∧ 13 = a * b + b * ((13 - a * b) / (a + b)) + a * ((13 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (15 + 1), ∃ b ∈ Finset.range (15 + 1), 0 < a ∧ a < b ∧ b < (15 - a * b) / (a + b) ∧ 15 = a * b + b * ((15 - a * b) / (a + b)) + a * ((15 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (16 + 1), ∃ b ∈ Finset.range (16 + 1), 0 < a ∧ a < b ∧ b < (16 - a * b) / (a + b) ∧ 16 = a * b + b * ((16 - a * b) / (a + b)) + a * ((16 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (18 + 1), ∃ b ∈ Finset.range (18 + 1), 0 < a ∧ a < b ∧ b < (18 - a * b) / (a + b) ∧ 18 = a * b + b * ((18 - a * b) / (a + b)) + a * ((18 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (21 + 1), ∃ b ∈ Finset.range (21 + 1), 0 < a ∧ a < b ∧ b < (21 - a * b) / (a + b) ∧ 21 = a * b + b * ((21 - a * b) / (a + b)) + a * ((21 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (22 + 1), ∃ b ∈ Finset.range (22 + 1), 0 < a ∧ a < b ∧ b < (22 - a * b) / (a + b) ∧ 22 = a * b + b * ((22 - a * b) / (a + b)) + a * ((22 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (24 + 1), ∃ b ∈ Finset.range (24 + 1), 0 < a ∧ a < b ∧ b < (24 - a * b) / (a + b) ∧ 24 = a * b + b * ((24 - a * b) / (a + b)) + a * ((24 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (25 + 1), ∃ b ∈ Finset.range (25 + 1), 0 < a ∧ a < b ∧ b < (25 - a * b) / (a + b) ∧ 25 = a * b + b * ((25 - a * b) / (a + b)) + a * ((25 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (28 + 1), ∃ b ∈ Finset.range (28 + 1), 0 < a ∧ a < b ∧ b < (28 - a * b) / (a + b) ∧ 28 = a * b + b * ((28 - a * b) / (a + b)) + a * ((28 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (30 + 1), ∃ b ∈ Finset.range (30 + 1), 0 < a ∧ a < b ∧ b < (30 - a * b) / (a + b) ∧ 30 = a * b + b * ((30 - a * b) / (a + b)) + a * ((30 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (33 + 1), ∃ b ∈ Finset.range (33 + 1), 0 < a ∧ a < b ∧ b < (33 - a * b) / (a + b) ∧ 33 = a * b + b * ((33 - a * b) / (a + b)) + a * ((33 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (37 + 1), ∃ b ∈ Finset.range (37 + 1), 0 < a ∧ a < b ∧ b < (37 - a * b) / (a + b) ∧ 37 = a * b + b * ((37 - a * b) / (a + b)) + a * ((37 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (40 + 1), ∃ b ∈ Finset.range (40 + 1), 0 < a ∧ a < b ∧ b < (40 - a * b) / (a + b) ∧ 40 = a * b + b * ((40 - a * b) / (a + b)) + a * ((40 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (42 + 1), ∃ b ∈ Finset.range (42 + 1), 0 < a ∧ a < b ∧ b < (42 - a * b) / (a + b) ∧ 42 = a * b + b * ((42 - a * b) / (a + b)) + a * ((42 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (45 + 1), ∃ b ∈ Finset.range (45 + 1), 0 < a ∧ a < b ∧ b < (45 - a * b) / (a + b) ∧ 45 = a * b + b * ((45 - a * b) / (a + b)) + a * ((45 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (48 + 1), ∃ b ∈ Finset.range (48 + 1), 0 < a ∧ a < b ∧ b < (48 - a * b) / (a + b) ∧ 48 = a * b + b * ((48 - a * b) / (a + b)) + a * ((48 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (57 + 1), ∃ b ∈ Finset.range (57 + 1), 0 < a ∧ a < b ∧ b < (57 - a * b) / (a + b) ∧ 57 = a * b + b * ((57 - a * b) / (a + b)) + a * ((57 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (58 + 1), ∃ b ∈ Finset.range (58 + 1), 0 < a ∧ a < b ∧ b < (58 - a * b) / (a + b) ∧ 58 = a * b + b * ((58 - a * b) / (a + b)) + a * ((58 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (60 + 1), ∃ b ∈ Finset.range (60 + 1), 0 < a ∧ a < b ∧ b < (60 - a * b) / (a + b) ∧ 60 = a * b + b * ((60 - a * b) / (a + b)) + a * ((60 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (70 + 1), ∃ b ∈ Finset.range (70 + 1), 0 < a ∧ a < b ∧ b < (70 - a * b) / (a + b) ∧ 70 = a * b + b * ((70 - a * b) / (a + b)) + a * ((70 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (72 + 1), ∃ b ∈ Finset.range (72 + 1), 0 < a ∧ a < b ∧ b < (72 - a * b) / (a + b) ∧ 72 = a * b + b * ((72 - a * b) / (a + b)) + a * ((72 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (78 + 1), ∃ b ∈ Finset.range (78 + 1), 0 < a ∧ a < b ∧ b < (78 - a * b) / (a + b) ∧ 78 = a * b + b * ((78 - a * b) / (a + b)) + a * ((78 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (85 + 1), ∃ b ∈ Finset.range (85 + 1), 0 < a ∧ a < b ∧ b < (85 - a * b) / (a + b) ∧ 85 = a * b + b * ((85 - a * b) / (a + b)) + a * ((85 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (88 + 1), ∃ b ∈ Finset.range (88 + 1), 0 < a ∧ a < b ∧ b < (88 - a * b) / (a + b) ∧ 88 = a * b + b * ((88 - a * b) / (a + b)) + a * ((88 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (93 + 1), ∃ b ∈ Finset.range (93 + 1), 0 < a ∧ a < b ∧ b < (93 - a * b) / (a + b) ∧ 93 = a * b + b * ((93 - a * b) / (a + b)) + a * ((93 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (102 + 1), ∃ b ∈ Finset.range (102 + 1), 0 < a ∧ a < b ∧ b < (102 - a * b) / (a + b) ∧ 102 = a * b + b * ((102 - a * b) / (a + b)) + a * ((102 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (105 + 1), ∃ b ∈ Finset.range (105 + 1), 0 < a ∧ a < b ∧ b < (105 - a * b) / (a + b) ∧ 105 = a * b + b * ((105 - a * b) / (a + b)) + a * ((105 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (112 + 1), ∃ b ∈ Finset.range (112 + 1), 0 < a ∧ a < b ∧ b < (112 - a * b) / (a + b) ∧ 112 = a * b + b * ((112 - a * b) / (a + b)) + a * ((112 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (120 + 1), ∃ b ∈ Finset.range (120 + 1), 0 < a ∧ a < b ∧ b < (120 - a * b) / (a + b) ∧ 120 = a * b + b * ((120 - a * b) / (a + b)) + a * ((120 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (130 + 1), ∃ b ∈ Finset.range (130 + 1), 0 < a ∧ a < b ∧ b < (130 - a * b) / (a + b) ∧ 130 = a * b + b * ((130 - a * b) / (a + b)) + a * ((130 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (133 + 1), ∃ b ∈ Finset.range (133 + 1), 0 < a ∧ a < b ∧ b < (133 - a * b) / (a + b) ∧ 133 = a * b + b * ((133 - a * b) / (a + b)) + a * ((133 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (165 + 1), ∃ b ∈ Finset.range (165 + 1), 0 < a ∧ a < b ∧ b < (165 - a * b) / (a + b) ∧ 165 = a * b + b * ((165 - a * b) / (a + b)) + a * ((165 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (168 + 1), ∃ b ∈ Finset.range (168 + 1), 0 < a ∧ a < b ∧ b < (168 - a * b) / (a + b) ∧ 168 = a * b + b * ((168 - a * b) / (a + b)) + a * ((168 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (177 + 1), ∃ b ∈ Finset.range (177 + 1), 0 < a ∧ a < b ∧ b < (177 - a * b) / (a + b) ∧ 177 = a * b + b * ((177 - a * b) / (a + b)) + a * ((177 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (190 + 1), ∃ b ∈ Finset.range (190 + 1), 0 < a ∧ a < b ∧ b < (190 - a * b) / (a + b) ∧ 190 = a * b + b * ((190 - a * b) / (a + b)) + a * ((190 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (210 + 1), ∃ b ∈ Finset.range (210 + 1), 0 < a ∧ a < b ∧ b < (210 - a * b) / (a + b) ∧ 210 = a * b + b * ((210 - a * b) / (a + b)) + a * ((210 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (232 + 1), ∃ b ∈ Finset.range (232 + 1), 0 < a ∧ a < b ∧ b < (232 - a * b) / (a + b) ∧ 232 = a * b + b * ((232 - a * b) / (a + b)) + a * ((232 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (240 + 1), ∃ b ∈ Finset.range (240 + 1), 0 < a ∧ a < b ∧ b < (240 - a * b) / (a + b) ∧ 240 = a * b + b * ((240 - a * b) / (a + b)) + a * ((240 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (253 + 1), ∃ b ∈ Finset.range (253 + 1), 0 < a ∧ a < b ∧ b < (253 - a * b) / (a + b) ∧ 253 = a * b + b * ((253 - a * b) / (a + b)) + a * ((253 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (273 + 1), ∃ b ∈ Finset.range (273 + 1), 0 < a ∧ a < b ∧ b < (273 - a * b) / (a + b) ∧ 273 = a * b + b * ((273 - a * b) / (a + b)) + a * ((273 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (280 + 1), ∃ b ∈ Finset.range (280 + 1), 0 < a ∧ a < b ∧ b < (280 - a * b) / (a + b) ∧ 280 = a * b + b * ((280 - a * b) / (a + b)) + a * ((280 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (312 + 1), ∃ b ∈ Finset.range (312 + 1), 0 < a ∧ a < b ∧ b < (312 - a * b) / (a + b) ∧ 312 = a * b + b * ((312 - a * b) / (a + b)) + a * ((312 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (330 + 1), ∃ b ∈ Finset.range (330 + 1), 0 < a ∧ a < b ∧ b < (330 - a * b) / (a + b) ∧ 330 = a * b + b * ((330 - a * b) / (a + b)) + a * ((330 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (345 + 1), ∃ b ∈ Finset.range (345 + 1), 0 < a ∧ a < b ∧ b < (345 - a * b) / (a + b) ∧ 345 = a * b + b * ((345 - a * b) / (a + b)) + a * ((345 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (357 + 1), ∃ b ∈ Finset.range (357 + 1), 0 < a ∧ a < b ∧ b < (357 - a * b) / (a + b) ∧ 357 = a * b + b * ((357 - a * b) / (a + b)) + a * ((357 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (385 + 1), ∃ b ∈ Finset.range (385 + 1), 0 < a ∧ a < b ∧ b < (385 - a * b) / (a + b) ∧ 385 = a * b + b * ((385 - a * b) / (a + b)) + a * ((385 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (408 + 1), ∃ b ∈ Finset.range (408 + 1), 0 < a ∧ a < b ∧ b < (408 - a * b) / (a + b) ∧ 408 = a * b + b * ((408 - a * b) / (a + b)) + a * ((408 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (462 + 1), ∃ b ∈ Finset.range (462 + 1), 0 < a ∧ a < b ∧ b < (462 - a * b) / (a + b) ∧ 462 = a * b + b * ((462 - a * b) / (a + b)) + a * ((462 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (520 + 1), ∃ b ∈ Finset.range (520 + 1), 0 < a ∧ a < b ∧ b < (520 - a * b) / (a + b) ∧ 520 = a * b + b * ((520 - a * b) / (a + b)) + a * ((520 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (760 + 1), ∃ b ∈ Finset.range (760 + 1), 0 < a ∧ a < b ∧ b < (760 - a * b) / (a + b) ∧ 760 = a * b + b * ((760 - a * b) / (a + b)) + a * ((760 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (840 + 1), ∃ b ∈ Finset.range (840 + 1), 0 < a ∧ a < b ∧ b < (840 - a * b) / (a + b) ∧ 840 = a * b + b * ((840 - a * b) / (a + b)) + a * ((840 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (1320 + 1), ∃ b ∈ Finset.range (1320 + 1), 0 < a ∧ a < b ∧ b < (1320 - a * b) / (a + b) ∧ 1320 = a * b + b * ((1320 - a * b) / (a + b)) + a * ((1320 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (1365 + 1), ∃ b ∈ Finset.range (1365 + 1), 0 < a ∧ a < b ∧ b < (1365 - a * b) / (a + b) ∧ 1365 = a * b + b * ((1365 - a * b) / (a + b)) + a * ((1365 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (1848 + 1), ∃ b ∈ Finset.range (1848 + 1), 0 < a ∧ a < b ∧ b < (1848 - a * b) / (a + b) ∧ 1848 = a * b + b * ((1848 - a * b) / (a + b)) + a * ((1848 - a * b) / (a + b)) All goals completed! 🐙

Idoneal numbers completeness conjecture.

@[category research open, AMS 11] theorem idoneal_numbers_completeness : answer(sorry) ↔ ∀ n : ℕ, IsIdoneal n → n ∈ knownIdonealNumbers := ⊢ True ↔ ∀ (n : ℕ), IsIdoneal n → n ∈ knownIdonealNumbers All goals completed! 🐙end Idoneal