1. FormalConjectures
  2. ErdosProblems
  3. «403»
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Erdős Problem 403

References:

    erdosproblems.com/403

    [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique (1980).

    [Li76] Lin, S., On two problems of Erdős concerning sums of distinct factorials. Bell Laboratories internal memorandum (1960).

namespace Erdos403

Does the equation $$2^m=a_1!+\cdots+a_k!$$ with $a_1<a_2<\cdots <a_k$ have only finitely many solutions?

Asked by Burr and Erdős. Frankl and Lin [Li76] independently showed that the answer is yes, and the largest solution is $$2^7=2!+3!+5!.$$ In fact Lin showed that the largest power of $2$ which can divide a sum of distinct factorials containing $2$ is $2^{254}$, and that there are only 5 solutions to $3^m=a_1!+\cdots+a_k!$ (when $m=0,1,2,3,6$).

See also [404].

A solution is encoded below as a pair $(m, s)$ where $s$ is the finite set ${a_1 < a_2 < \cdots < a_k}$ of positive integers, so the distinctness of the $a_i$ is given by set membership. The empty set contributes no solutions since $2^m \geq 1 > 0$.

@[category research solved, AMS 11] theorem declaration uses 'sorry'erdos_403 : answer(True) ↔ {p : ℕ × Finset ℕ | (∀ a ∈ p.2, 0 < a) ∧ 2 ^ p.1 = ∑ a ∈ p.2, a.factorial}.Finite := ⊢ True ↔ {p | (∀ a ∈ p.2, 0 < a) ∧ 2 ^ p.1 = ∑ a ∈ p.2, a.factorial}.Finite All goals completed! 🐙 end Erdos403