Other Number Theory

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E0034/: Asks whether every permutation of the first n integers has only o(n squared) distinct sums of consecutive terms; false, by Hegyvári's 1986 construction and Konieczny's explicit permutation with at least n squared over 4 distinct sums, the maximum lying between 0.286 and 0.446 times n squared.

E0268/: Asks whether the triples of reciprocal sums over n, n plus one and n plus two, taken over infinite sets with convergent reciprocal sum, contain an open set.

E0362/: Bounds the number of subsets of an N-element set of naturals summing to a fixed target by 2^N over N^{3/2}, proved by Sárközy and Szemerédi with Stanley's exact maximizers, and the count with the subset size also fixed by 2^N over N squared, proved by Halász from his bound on signed sums of 1-separated plane vectors in a unit ball.

E0464/: Asks whether every lacunary sequence admits an irrational multiplier whose fractional parts along the sequence are not dense in the unit interval; proved by Pollington and de Mathan, with Peres and Schlag's separation of order epsilon over log(1/epsilon), while the site's wording is trivially true.

E0465/: Asks whether the largest set of points in a disc of radius X whose pairwise distances all stay at least delta from the integers has o(X) points, and even fewer than X to the one half plus o(1); proved by Sárközy (the first bound) and Konyagin (the sharp exponent one half).

E0466/: Asks whether for some fixed delta the largest set of points in a disc of radius X whose pairwise distances all stay at least delta from the integers grows without bound as X grows; the corrected statement takes the limit in X, which the site misprints, and Sárközy's power lower bound proves it.

E0480/: Asks whether every sequence in [0,1] has a gap n for which the lower limit of n times the spacing of terms n apart is at most one over root five; proved by Chung and Graham with the sharp constant 0.3944....

E0482/: Asks for analogs, for root m and other algebraic numbers, of the Graham-Pollak recurrence whose differences a_{2n+1} - 2a_{2n-1} are the binary digits of root two; solved by Stoll's families for every positive real and every base.

E0492/: Asks whether, for a real sequence tending to infinity whose consecutive ratios tend to one, the positions of the multiples of almost every real within the sequence's gaps are uniformly distributed; LeVeque's question, disproved by Schmidt. The site's wording restricts to integer sequences, for which the Davenport–Erdős theorem gives yes.

E0951/: Asks whether a sequence of reals whose distinct integer power products always differ by at least 1 has no more terms up to x than there are primes up to x; the quantifier over x is implicit, and the two readings are parts.

E0952/: Asks whether there is an infinite sequence of distinct Gaussian primes in which consecutive terms are always a bounded distance apart; the Gaussian moat problem, answered negatively by an accepted 2026 claim with a Lean proof.

E0963/: Estimates the largest dissociated subset guaranteed in any set of n reals, in particular whether it always has at least floor(log_2 n) elements.

E0972/: Asks whether, for irrational alpha > 1, infinitely many primes p have the integer part of p alpha prime; open, the one-prime statement classical and the two-prime statement known only for almost all alpha.

E0981/: Asks whether the sum over primes below x of the eventual-time threshold of the Legendre-symbol partial sums is roughly x over log x; proved by Elliott (1969) for the two-sided threshold, the one-sided form left to his remark.

E1005/: Estimates f(n), the largest d such that Farey fractions of order n at most d places apart are similarly ordered; asks if f(n) = (c + o(1))n. Marked solved on Cipollini's 2026 preprint (c = 1/4, matching van Doorn's upper bound).

E1096/: Asks whether the gaps between consecutive finite sums of distinct powers of q tend to zero for every q slightly above one; proved by Erdős and Komornik (1998), Akiyama and Komornik (2013) and Feng (2016), each for a range of q.

E1135/: Asks whether every orbit of the shortcut Collatz map reaches one; open, with the site's caveat on the reported Erdős prize figure, verification below 2^71, no cycle with at most 91 local minima, and Tao's almost-all theorem.

E1180/: Asks whether, for each positive epsilon, boundedly many inverses of integers up to p^epsilon represent every residue modulo any prime p; proved by Shparlinski, Croot and Glibichuk, whose bound has order epsilon^(-2).

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The general number theory problems fitting none of the three themed folders, including questions about real numbers, reciprocal sums as points, irrational rotations and floor sequences, quadratic residues and character sums, and other one-off constructions.

Site tags routed here: iterated functions, number theory.