Erdős problem 399 · problems.science

Is it true that there are no solutions to n! = x^k ± y^k with x,y,n ∈ ℕ, x*y > 1, and k > 2?

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5 retained statements2415f78e850a

FormalConjectures/ErdosProblems/

399.lean

Retained formal statement5 of 5

Erdős and Obláth observed that the Bertrand-style fact (first proved by Breusch [Br32]) that, if $q_i$ is the sequence of primes congruent to $3\pmod{4}$ then $q_{i+1}<2q_i$ except for $q_1=3$ , together with Fermat's theorem on the sums of two squares implies that the only solution to $n!=x^2+y^2$ is $6!=12^2+24^2$ .

FormalConjectures/ErdosProblems/399.leanErdos399.erdos_399.variants.sum_two_squares1 lineExact file

1∀ {n x y : ℕ}, 1 < x * y → n.factorial = x ^ 2 + y ^ 2 → n = 6 ∧ (x = 12 ∧ y = 24 ∨ x = 24 ∧ y = 12)

SolvedStatement only, no proof

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