Problems

Define φ_k(n) = ∑_1 ≤ a ≤ n, (a,n)=1 a^k and D_s = k ≥ s : φ_s(n) | φ_k(n) for every n. Is D_1 = 1, 3, 15, as conjectured by Büyükaşik and collaborators?

Let a,b,c>1 be pairwise coprime integers. Is every large integer a sum of distinct numbers of the form a^k b^l c^m (k,l,m≥ 0), none dividing another?

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

Banks and Martin conjectured in 2013 that for a primitive set A and any set Q of primes, the Erdos sum of the members of A composed only of primes in Q is at most the corresponding sum over Q itself. The unrestricted form turned out to be…

Whether there are infinitely many integers a, b, n with a, b ≥ ε n such that a!· b! divides n!·(a+b-n)! while a+b exceeds n by more than C·log n.

Is the maximum size of a set A⊆ 1,…,N such that ab+1 is never squarefree (for all a,b∈ A) achieved by taking those n≡ 7pmod25? Resolved for all sufficiently large N: any near-maximal A is contained in n≡ 7pmod25 or n≡ 18pmod25, leaving…

Let F(n) be the largest A⊆1,…,n with anmid bc for distinct a,b,c∈ A. Is F(n)=π(n)+(C+o(1)) n^2/3(log n)^-2 for some constant C?

Nathanson asked which subsets of N can occur as product intersection sets of a family of semigroup subsets, for arbitrary and for decreasing families (his Problems 10 and 11). Both are solved by complete classifications.

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

For every real ξ>0 the sequence of integer parts [ξ 7^n], n=0,1,2,…, contains infinitely many composite numbers. Second, there is no infinite right truncatable prime in base~7.

Let k,r≥ 2. Does there exist a set A⊆ N that contains no non-trivial arithmetic progression of length k+1, yet in any r-colouring of A there must exist a monochromatic non-trivial arithmetic progression of length k? Answered in the…

VibeMathed records no statement for this problem. See erdosproblems.com for the original.