Erdős Problem #1108
For each k ≥ 2, does the set A = Σ_n∈ Sn! : S⊂ ℕ finite of all finite sums of distinct factorials contain only finitely many k-th powers?
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Open problems about factorials
Equations and divisibility questions involving n!, products of consecutive integers, and their prime factors.
Erdős Problem #18
Conjecture 1. Are there infinitely many practical numbers m such that h(m) < (log log m)^O(1)? More precisely: does there exist a constant C > 0 such that for infinitely many practical numbers m, we have h(m) < (log log m)^C?
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Erdős Problem #373
Show that the equation n!=a_1!a_2!···a_k!, with n−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k, has only finitely many solutions.
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Erdős Problem #390
Does there exists a constant c such that f n - 2 n ~ c (n / log n)?
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Erdős Problem #398
Brocard's Problem Does n! + 1 = m^2 have integer solutions other than n = 4, 5, 7?
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Erdős Problem #400
Can one show that Σ_n≤ xg_k(n) ∼ c_k xlog x for some constant c_k?
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Erdős Problem #478
Let p be a prime and A_p = k! pmodp : 1≤ k<p. Is it true that lvert A_prvert ∼ (1-1/e)p?
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Erdős Problem #727
Let k ≥ 2. Does ((n+k)!)^2∣(2n)! hold for infinitely many n?
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Erdős Problem #912
Prove that there exists some c>0 such that h(n) ∼ c (n/log n)^1/2 as n→ ∞.
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