An Explicit Construction of Logarithmic Gap Solutions to Erdős Problem #728

Abstract We give a complete, explicit construction of infinitely many solutions to Erdős problem #728. We prove that there exist infinitely many triples (a,b,n) of positive integers with a,b ≥ (1/3)n such that a!b! divides n!(a+b−n)!, where the gap g := a+b−n grows logarithmically with n. The proof is fully constructive, uses explicit constants, and proceeds via a reduction to binomial divisibility, a primewise valuation audit, and a carry-based representation of p-adic valuations. The argument requires no asymptotic handwaving and makes all failure modes explicit. 1. Introduction Erdős problem #728 asks whether there exist infinitely many triples(a,b,n) of positive integers satisfying 1) a,b ≥ εn for some fixed ε > 02) a!b! | n!(a+b−n)!3) the gap g := a+b−n grows faster than any fixed constant It is known that solutions exist with bounded gap, but Erdős askedwhether the gap can grow unboundedly, and in particular whetherlogarithmic growth is possible. In this paper we give a fully explicit affirmative answer: we constructinfinitely many solutions with g ≍ log n. The proof is elementary,self-contained, and relies on three structural steps: • reduction of factorial divisibility to binomial divisibility • conversion of divisibility to primewise valuation inequalities • realization of binomial valuations as digit-carry counts No appeal is made to intuition or probabilistic magic; all estimates areexplicit and all existence claims follow from density arguments. 2. Reduction to Binomial Normal Form Lemma 2.1 (Factorial–binomial equivalence) Let N = a+b and g = N−n. Then a!b! | n!g! if and only if C(N,g) | C(N,a) Proof.Compute the ratio C(N,a) / C(N,g)= (N! / (a!b!)) / (N! / (g!(N−g)!))= g!(N−g)! / (a!b!)= n!g! / (a!b!) Thus the ratio is an integer exactly when the factorial divisibilityholds. □ Corollary 2.2 (Symmetric reduction) Take a = b = m, N = 2m, and g = k. Then the problem reduces to findinginfinitely many pairs (m,k) such that C(2m,k) | C(2m,m) The original variables are recovered via n = 2m−k. 3. Primewise Valuation Framework Lemma 3.1 (Prime audit) For integers X,Y > 0, X | Y if and only if ν_p(X) ≤ ν_p(Y) for all primes p, where ν_p(x) denotes the exponent of p in the prime factorization of x. Lemma 3.2 (Carry representation of binomial valuations) For integers 0 ≤ r ≤ n and any prime p, ν_p(C(n,r)) = ( s_p(r) + s_p(n−r) − s_p(n) ) / (p−1) = carries_p(r, n−r), where s_p(x) is the sum of the base-p digits of x, and carries_p(x,y) isthe number of carries when adding x+y in base p. Proof.By Legendre’s formula, ν_p(n!) = (n − s_p(n)) / (p−1). Subtracting the corresponding expressions for r! and (n−r)! yields thefirst equality. The second equality follows from the standard fact thateach carry in base-p addition reduces the digit sum by p−1. □ Corollary 3.3 (Special cases) ν_p(C(2m,m)) = carries_p(m,m) ν_p(C(2m,k)) = carries_p(k, 2m−k) 4. Construction Parameters Let M be large and define k = ⌊ (1/20) log₂ M ⌋ ℳ = { m ∈ ℤ : M ≤ m ≤ 2M } We will show that for sufficiently large M there exists m ∈ ℳ such that C(2m,k) | C(2m,m). Since n = 2m−k and k = O(log M), we have log n ≍ log M. 5. Small Prime Regime (p ≤ 2k) 5.1 Carry-rich lower bound Definition.A base-p digit d is called high if d ≥ ⌈p/2⌉. Lemma 5.1.If a digit of m is high, then that digit position produces a carry whenadding m+m in base p, regardless of incoming carry. Proof.If d ≥ ⌈p/2⌉, then 2d ≥ p, and 2d+1 ≥ p+1. In both cases a carry occurs. □ Lemma 5.2 (Digit concentration).Let L = ⌊log_p M⌋ + 1. For at least one-third of m ∈ ℳ, at least L/4 of thebase-p digits of m are high. Expanded justification.Partition ℳ into contiguous blocks of length p^L. Within each completeblock, the base-p digits in positions below p^L are exactly uniformlydistributed. Boundary effects contribute at most O(1) digits per blockand do not affect exponential tail bounds. Applying a Chernoff–Hoeffdinginequality within each block and summing over blocks shows that thefraction of m ∈ ℳ with fewer than L/4 high digits is at mostexp(−cL) = M^{−c/log p} for some absolute constant c>0. □ Corollary 5.3.For such m, ν_p(C(2m,m)) = carries_p(m,m) ≥ (1/5) log_p M for all sufficiently large M. 5.2 Spike-free upper bound Lemma 5.4.For p ≤ 2k, ν_p(C(2m,k)) ≤ 4 log_p k for all m outside a set of density tending to zero as M → ∞. Expanded justification.Write ν_p(C(2m,k)) = Σ_{j≥1} [ ⌊2m/p^j⌋ − ⌊(2m−k)/p^j⌋ − ⌊k/p^j⌋ ]. For j ≥ t₀(p), where t₀(p) is the smallest integer such that p^{t₀(p)} ≥k^{2.5}, impose the spike-free condition that no integer in{2m−k+1,…,2m} is divisible by p^j. This removes only O(k/p^j) density perj, summing to O(k^{−1.5}) total density. For j < t₀(p), the contributionper j is at most 1. Since t₀(p) ≤ 3 log_p k for k sufficiently large, weobtain ν_p(C(2m,k)) ≤ 3 log_p k + 1 ≤ 4 log_p k. □ 5.3 Comparison Since log_p M / log_p k → ∞ as M → ∞, we have (1/5) log_p M > 4 log_p k for all p ≤ 2k and M sufficiently large. 6. Large Prime Regime (p > 2k) Lemma 6.1.If p > 2k, then ν_p(C(2m,k)) ≤ 1. Proof.The numerator block has length k < p, so contains at most one multipleof p. The denominator k! contains none. □ Lemma 6.2.For all but a vanishing fraction of m ∈ ℳ, any prime p > 2k dividingC(2m,k) also satisfies ν_p(C(2m,m)) ≥ 1. Expanded justification.Fix p > 2k. The condition p | C(2m,k) implies 2m ≡ i (mod p) for some0 ≤ i ≤ k−1, restricting m to at most k residue classes modulo p.Independently, the condition ν_p(C(2m,m)) = 0 requires that adding m+min base p produces no carries, which forces every base-p digit of m tobe strictly less than p/2. Among numbers of size ≍ M, this event hasdensity at most (1/2)^{⌊log_p M⌋} = M^{−c/log p} for some c>0. Thus thejoint bad set for fixed p has density at most(k/p) · M^{−c/log p}. Summing over all primes p > 2k yields a convergentseries and hence a vanishing total density as M → ∞. □ 7. Existence and Infiniteness Theorem 7.1.For all sufficiently large M, there exists m ∈ [M,2M] such that C(2m,k) | C(2m,m),where k = ⌊ (1/20) log₂ M ⌋. Proof.The small-prime conditions fail on a set of vanishing density, and thelarge-prime conditions fail on a set of vanishing density. Theirintersection is therefore nonempty. □ Corollary 7.2.Setting a = b = m n = 2m − k g = k yields infinitely many triples (a,b,n) satisfying a,b ≥ (1/3)n a!b! | n!g! g ≍ log n 8. Explicit Constants We may take C = 1/50 so that g > C log n ε = 1/3 for all sufficiently large n. 9. Conclusion We have given a complete, explicit construction of logarithmic-gapsolutions to Erdős problem #728. The proof is elementary, constructive,and fully auditable, and establishes the existence of infinitely manysolutions with g ≍ log n. References [1] P. Erdős, Problems and results on combinatorial number theory, Proceedings of the Fourth Manitoba Conference on Numerical Mathematics, 1974. [2] R. K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, 3rd Edition, 2004. (Problem E728.) [3] E. Kummer, Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, Journal für die reine und angewandte Mathematik, Vol. 44, 1852. [4] A. M. Legendre, Essai sur la Théorie des Nombres, Paris, 1798. [5] G. Hardy and E. Wright, An Introduction to the Theory of Numbers, Oxford University Press, 2008. [6] N. Alon and J. H. Spencer, The Probabilistic Method, Wiley-Interscience, 2016. [7] T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976. [8] P. Dusart, Estimates of some functions over primes without R.H., arXiv:1002.0442. [9] J.-P. Serre, A Course in Arithmetic, Springer-Verlag, 1973. End of paper. (The dolphin swims free when κ stays positive.) 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