Problems
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For A ⊆ F_p let A^* = (A+A) ∪ (AA). Sárközy conjectured that for all large primes, every set of size at least c√p has A^* = F_p-like covering behaviour. Disproved with an explicit construction from the classical cross-ratio orbit, together…
For the Erdős–Pomerance functions F(n) and h_P(n) counting how many consecutive integers are needed to contain a distinct multiple of each integer, respectively prime, up to n, the paper proves F(n) ≥ h_P(n) ≥ nexp((log 2/2 - o(1))log…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
22 conjectures of Cohen about cyclic numbers (integers with gcd(n, φ(n)) = 1) settled at once - 16 proved, 6 disproved - together with a complete resolution of a related OEIS problem on sequences whose running averages are Fibonacci…
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.
Let n_k be the least integer greater than 2k for which ∏_i=1^k (n_k - i) has no prime factor in (k, 2k). How rapidly must n_k grow?
The Riemann hypothesis asserts that every nontrivial zero of the zeta function lies on the critical line. Short of proving it, the standard measure of progress is the proportion of zeros known unconditionally to lie there: Selberg…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
Erdos and Graham asked whether a positive-density subset of 1,…,N can avoid having any two distinct elements a,b whose unit fractions average to a unit fraction. It can: there is a constant c>0 such that for all large N some A ⊆ 1,…,N of…
If a/b ∈ Q_>0 and b is squarefree, can a/b always be written as a finite sum of reciprocals of distinct products of two distinct primes?
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.
How long must an interval be to contain distinct representatives x_i, with a_i | x_i, for every n-element set of moduli A = a_1, …, a_n?
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…
Let h(n) count powerful integers in [n^2, (n+1)^2). What is the extremal order of h(n)?
Let a_1 = 2 and a_2 = 3 and continue the sequence by appending to a_1, …, a_n all possible values of a_ia_j - 1 with i ≠ j. Is it true that the set of integers which eventually appear has positive density?
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.
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.