Factorials and Binomial Coefficients

../

E0175/: Asks whether the central binomial coefficient of 2n choose n fails to be squarefree for every n at least 5; proved for large n by Sárközy (1985) and for every n at least 5 by Velammal (1995) and by Granville and Ramaré (1996).

E0373/: Asks whether a factorial can equal a product of two or more smaller factorials, each at least two, only finitely often.

E0376/: Asks whether infinitely many central binomial coefficients are coprime to one hundred and five.

E0377/: Asks whether the sum of the reciprocals of the primes up to n that do not divide the central binomial coefficient of n is bounded by a constant.

E0378/: Asks whether the integers n with at least r squarefree binomial coefficients in row n have a density, and whether that density is positive; both answered yes by Granville and Ramaré's 1996 theorem.

E0379/: Asks whether the largest exponent S such that every binomial coefficient in row n is divisible by some prime to the power S is unbounded as n varies; answered yes in 2025 by Cambie, Kovač and Tao.

E0384/: Asks whether, for 1 < k < n - 1, every binomial coefficient n choose k has a prime divisor at most n/2, except 7 choose 3; proved by Ecklund in 1969, while the site's strict bound p < n/2 fails at 4 choose 2.

E0386/: Asks whether a binomial coefficient, other than the trivial ones, can be a product of consecutive primes infinitely often.

E0387/: Asks whether some constant c makes every binomial coefficient in row n have a divisor between c times n and n.

E0390/: Asks whether the least top factor in a factorization of n factorial into increasing factors above n exceeds two n by about a constant times n over log n.

E0391/: Bounds the largest possible smallest factor when n factorial is written as a product of n increasing factors, in particular whether it approaches n over e.

E0392/: Asks whether the fewest factors needed to write n factorial as increasing factors of size at most n squared is about n over two minus n over two log n.

E0393/: Determines the behavior of the smallest spread between largest and smallest factor when a factorial is written as a product of distinct increasing integers.

E0396/: Asks whether, for every k, some n makes the product of the k plus one integers from n minus k to n divide the central binomial coefficient of n.

E0397/: Asks whether only finitely many equalities hold between two products of central binomial coefficients taken over distinct indices.

E0398/: Asks whether a factorial is one less than a perfect square only for n equal to four, five, and seven.

E0399/: Asks whether a factorial can equal a sum or difference of two kth powers with k greater than two and the powers not both trivial.

E0400/: Asks for the average and typical size of the largest excess of a sum of numbers whose factorials divide n factorial over n, for each fixed count k of terms.

E0401/: Asks whether, for infinitely many n, two numbers whose factorials' product divides n! times the n-th power of the product of the first r primes can sum to more than n plus f(r) log n, with f(r) tending to infinity.

E0403/: Asks whether a power of two can equal a sum of distinct factorials in only finitely many ways.

E0404/: Asks for which integers a and primes p the power of p dividing a sum of increasing factorials starting at a is bounded, and how that bound behaves.

E0405/: Asks whether, for each odd prime p, the equation with p minus one factorial plus a power of a equal to a power of p has only finitely many solutions.

E0419/: The set of limit points of the ratio of the number of divisors of n plus one factorial to the number of divisors of n factorial.

E0478/: Asks whether the number of distinct factorial residues modulo a prime is asymptotically one minus one over e times the prime.

E0646/: Asks whether, for any finitely many distinct primes, infinitely many n make n factorial divisible by an even power of each of those primes.

E0683/: Asks whether the largest prime divisor of n choose k is always at least the smaller of n minus k plus 1 and k to a power greater than one.

E0684/: Splits n choose k into the part made of primes at most k and the part made of primes above k, and asks how the sizes of the two parts compare.

E0685/: Asks whether, for k in a middle range, the distinct prime divisors of n choose k number about k times the sum of one over p for primes p between k and n.

E0698/: Asks whether the greatest common divisor of n choose i and n choose j always tends to infinity with n, uniformly over all i and j between 2 and half of n.

E0699/: Asks whether for all i less than j up to half of n some prime at least i divides both n choose i and n choose j.

E0700/: Studies the least greatest common divisor of n and n choose k over k between 1 and half of n, asking when it is large and how big it can be for composite n.

E0727/: Asks, for each k at least 2, whether the square of the factorial of n plus k divides the factorial of two n for infinitely many n.

E0728/: Asks whether infinitely many triples a, b, n with a plus b above n plus C log n have a! b! dividing n! (a+b-n)!; answered yes in 2026 by an AI-generated proof credited to Barreto and, separately, by Pomerance.

E0729/: Asks whether infinitely many triples a, b, n have a plus b above n plus C log n while n! over a! b! has only bounded primes in its denominator; answered yes in 2026 by an AI-generated proof credited to Barreto and Price.

E0730/: Asks whether infinitely many pairs of distinct integers n and m give central binomial coefficients with exactly the same set of prime divisors; answered yes in 2026, with consecutive pairs, by an AI proof credited to Price.

E0731/: Asks for a function describing, for almost all n, the least integer that fails to divide the central binomial coefficient of n.

E0849/: Asks whether, for every t at least one, some value is taken by exactly t binomial coefficients with the lower index between one and half the upper.

E0912/: Estimates how many distinct exponents occur in the prime factorization of n factorial.

E1093/: Studies the deficiency of n choose k, the number of the k integers from n downwards whose prime factors are all at most k, when no prime up to k divides it.

E1094/: Asks whether the least prime factor of n choose k is at most the larger of n over k and k for all n at least 2k, with only finitely many exceptions.

E1095/: Estimates the smallest n greater than k plus one for which every prime factor of n choose k exceeds k.

E1108/: Asks whether the set of sums of distinct factorials contains only finitely many k-th powers for k at least 2, and only finitely many powerful numbers.

---

Divisibility, prime factorization and digit arithmetic of factorials, binomial coefficients, and products of consecutive integers.

Site tags routed here: base representations, binomial coefficients, factorials, number theory, primes.