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Problem 699

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claims/: The 2 claim pages of Problem 699, one per claimant's result; the problem's standing derives from them.

Statement. Is it true that for every 1≤i<j≤n/21\leq i<j\leq n/2 there exists some prime p≥ip\geq i such that

p∣gcd((ni),(nj))?p\mid \textrm{gcd}\left(\binom{n}{i}, \binom{n}{j}\right)?

Status. Falsifiable: the site's label, which says the question is open and a single triple would refute it. The site credits GPT 5.6, prompted by Price, with the cases j≤3i/2j\le 3i/2 and n=2jn=2j (page edited 19 July 2026); see the Price claim page. Van Doorn and Rocca's manuscripts, on their claim page, reduce the problem to i=3i=3 and a finite set and are not credited on the site. The standing in the frontmatter derives from the claim pages.

Source. erdosproblems.com/699, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #699, https://www.erdosproblems.com/699. The site attributes the problem to Erdős and Szekeres [ErSz78].

References.

  • [ErSz78] Erdős, P. and Szekeres, G., Some number theoretic problems on binomial coefficients. Austral. Math. Soc. Gaz. 5 (1978), 97-99; the conjecture is equation (4), p. 97. Library home: erdos_1978_number_theoretic_problems_binomial_coefficients.
  • [Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp. Section B31 "Binomial coefficients", printed p. 131: the noncoprimality of (nr)\binom nr and (ns)\binom ns for 0<r<s≤n/20<r<s\le n/2 and the Erdős--Szekeres question whether the greatest prime factor of the g.c.d. is always greater than rr, with one noticed counterexample for r>3r>3; the identity and the counterexample are displayed formulas not reproduced here. Library home: guy_2004_unsolved_problems_number_theory.

Formalization. Statement in formal-conjectures.

Current assessment

Van Doorn and Rocca's two unpublished 2026 manuscripts, Partial Progress on Erdős Problem #699 and Binomial coefficients sharing a large prime divisor, settle i=1,2i=1,2, exclude every i≥1476i\ge1476, and prove finiteness for each fixed i≥4i\ge4. The first manuscript says that all results and arguments specific to its solution follow L. Price's Overleaf project Common Prime Divisor of Binomial Coefficients (2026). The manuscripts' proofs have not been reviewed here; the full problem remains unresolved. The site's falsifiable label means that a single bad triple (n,i,j)(n,i,j), checked by finite arithmetic, would refute the statement; the finiteness results above bound where such a triple could lie but do not exhibit one. This page records no literature-status search supporting the site's falsifiable label beyond the site record and the sources named here.

Claims. The two results claimed about the problem have claim pages, from which the frontmatter standing derives. Price's partial claim of 2026-07-18, on the Price claim page, proves the cases j≤3i/2j\le3i/2 and n=2jn=2j with GPT 5.6 Sol Pro and is pending: the site credits it, but on a problem it labels FALSIFIABLE. Van Doorn and Rocca's partial claim of 2026-07-25, on the van Doorn–Rocca claim page, is the forum posting of the first manuscript above, with the second added to the same Overleaf project later, and is pending. Neither settles the problem, and no claim settles the case i=3i=3 (both cover only its triples with j=4j=4, and Price's also those with n=2jn=2j). A proof posted to the site's thread on 2026-04-30, since deleted, was judged invalid by the curator on 2026-05-11, who also found that its Lean code did not formalize the statement; as a deleted thread post it has no claim page.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (63)
Problem 698Price's GPT 5.6 Sol Pro proof for j at most 3i/2 and for n = 2jVan Doorn and Rocca's reduction to i = 3 and a finite setproblems/factorials_binomials/E0699/claimsFactorials and Binomial CoefficientsErdős 699: current results and remaining questionsFactorials and Binomial Coefficientsfactorials_binomials/bergman_2011_common_divisors_multinomial_coefficientsTheorem 1 (p. 1): two binomial coefficients N choose i and N choose j with 0 < i <= j <= N/2 have a common divisor greater than 1Theorem 2 (p. 3): for 2 <= i <= j <= N/2 the gcd of N choose i and N choose j is at least N^{1/2} 2^{i-7/2} / i(i-1)^{1/2}Corvaja–Zannier: Integral points, divisibility between values of polynomials and entire curves on surfacesLemma 1 (p. 10): a valuation inequality between local equations is integrality on the blow-upTheorem 2 (p. 4): three divisibilities f_i(x,y) | g_i(x,y) in general position have non-dense solutionsTheorem 6 (p. 6): for n >= 6 cyclic linear forms the ideal identities F_i (x,y,z) = (F_{i-1},F_i)(F_i,F_{i+1}) hold on a non-dense setCroot et al.: On a conjecture of Graham on the p-divisibility of central binomial coefficientsTheorem 1 (p. 3): for distinct primes p_1,...,p_r >= c_0(r,eps), infinitely many n have every nu_{p_i} of the central binomial coefficient at most eps log n / log p_iEstimates of Some Functions Over Primes without R.H.Proposition 6.8 (p. 8): for x >= 396738 there is a prime p with x < p <= x(1 + 1/(25 ln^2 x))Theorem 5.2 (p. 4): |theta(x) - x| < eta_k x / ln^k x for x >= x_k, with a table of constantsOn the prime factorization of binomial coefficientsCorollary (p. 259): finitely many n choose k with U > V, nineteen found, complete except for k = 3, 5, 7Main theorem (p. 258): the part of n choose k below k exceeds the part at least k in exactly 12 casesfactorials_binomials/erdos_1934_theorem_sylvester_schurTheorem (Sylvester–Schur): a block of k consecutive integers above k has a prime factor greater than kfactorials_binomials/erdos_1978_number_theoretic_problems_binomial_coefficientsConjecture 1 (p. 97): a common prime factor at least i of C(n,i) and C(n,j)Inequality 3 (p. 97): two binomial coefficients in one row have gcd at least 2^iArithmetic Properties of Binomial Coefficients I: Binomial Coefficients modulo Prime PowersHeier–Levin: A Schmidt-Nochka Theorem for closed subschemes in subgeneral positionLemma 3.1 (p. 9) and Corollary 3.3 (p. 12): a generalized Chebyshev inequality for decreasing a_i and nonnegative b_i, c_iTheorem 1.2 (p. 2): a weighted Schmidt subspace inequality for arbitrary closed subschemes, with coefficient (n+1) max alpha_v(W)/codim Wfactorials_binomials/li_2026_erdos_problem_684_at_density_oneRousseau et al.: Divisibility of polynomials and degeneracy of integral pointsLemma 2.4 (p. 6): integrality on a blow-up is a local-height inequality downstairsTheorem 1.1 (p. 2): divisibility of polynomial values at S-integral points in P^n is degenerateTheorem 1.7 (p. 3): points satisfying a cyclic system of ideal equalities lie in a Zariski closed setPrime factors of arithmetic progressions and binomial coefficientsSection 1.4 (p. 5): the greatest prime factor of a binomial coefficient exceeds 1.95k, and is at least of order k log k loglog k / logloglog kTheorem 1 (p. 6): lower bounds for the number of distinct prime factors of a binomial coefficientBinomial coefficients sharing a large prime divisorPartial Progress on Erdős Problem #699Lemma 2.1 (p. 2): for a bad triple, V_i(n) divides C(j, i) and C(n − j, i)Lemma 2.2 (p. 2): Kummer localization of q^a ∥ C(n, i) at a point of Δ_iLemma 4.1 (p. 4): a polynomial vanishing to order m on Δ_i gives V_i(n)^m | F(j, n − j)Proposition 2.4 (p. 3): every admissible triple with j ≤ 3i/2 is goodProposition 3.1 (p. 3): there are no bad triples with i = 1 or i = 2Proposition 4.2 (p. 4): for i ≥ 5 and a bad triple, V_i(n) < n^{ρ_i} with ρ_i < i − 1Proposition 4.3 (p. 5): for a bad triple with i = 4, V_4(n) < n^{17/6}Theorem 1.2 (p. 1): no bad triples for i = 1, 2 or i ≥ 1476, finitely many for 4 ≤ i ≤ 1475Theorem 4.4 (p. 6): for every fixed i ≥ 4 there are only finitely many bad triples
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