Erdős Problem #1093
Are there infinitely many binomial coefficients with deficiency 1?
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Open problems about binomial coefficients
Prime factors, divisibility and multiplicative structure of binomial coefficients. Statements are elementary, which makes them good targets for computation and for Lean formalisation.
Erdős Problem #1094
For all n≥ 2k the least prime factor of C(n, k) is ≤max(n/k,k), with only finitely many exceptions.
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Erdős Problem #376
Are there infinitely many n such that 2nchoose n is coprime to 105?
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Erdős Problem #377
Is there some absolute constant C > 0 such that Σ_p ≤ n 1_pnmid 2n choose n1/p ≤ C for all n?
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Erdős Problem #386
Let 2 ≤ k ≤ n - 2. Can C(n, k) be the product of consecutive primes infinitely often? Here k may vary with n: the question asks for infinitely many admissible binomial coefficients, not for a single k that works infinitely often.
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Erdős Problem #396
Is it true that for every k there exists n such that Π_0≤ i≤ k(n-i) | C(2n, n)?
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Erdős Problem #683
Let P(n, k) be the largest prime factor of C(n, k). There exists c > 0 such that P(n, k) ≥ min(n - k + 1, k^1 + c) for all 0 < k ≤ n/2. Erdős stated this for 1 ≤ k ≤ n with the bound min(n-k+1, k^1+c) [Er79d].
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Erdős Problem #699
Erdős Problem 699. Is it true that for every 1 ≤ i < j ≤ n / 2 there exists a prime p ≥ i with p | gcd(C(n, i), C(n, j))?
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Erdős Problem #700
Let f(n) = min_1 < k ≤ n/2 gcd(n, C(n, k)) and let P(n) be the largest prime dividing n. (a) Characterise those composite n such that f(n) = n/P(n). Erdős–Szekeres [ErSz78] note that f(n) = n/P(n) when n is a product of two primes (erdos_700.variants.prime_mul), with n = 30 a further example.
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Erdős Problem #849
Is it true that, for every integer t≥1, there is some integer a such that n choose k = a with 1≤ k ≤ n/2 has exactly t solutions?
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