Problems
Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
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0468
Problem 468
OpenStudies the set of partial sums of the increasing divisors of n above one.
- Number theory
- Divisors
Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
516 of 1,221 problems match
Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
Studies the set of partial sums of the increasing divisors of n above one.
Studies weird numbers, those whose divisor sum is at least twice the number yet which are not the sum of any set of their own divisors.
Asks whether some starting list of primes makes infinite the sequence whose next term is the least prime of the form previous plus an earlier term minus one.
Whether every finite set of nonzero residues modulo a prime can be ordered with all partial sums distinct (Graham's rearrangement conjecture); proved for all large primes by four range results with no explicit threshold.
Asks whether the number of distinct factorial residues modulo a prime is asymptotically one minus one over e times the prime.
Asks whether, for every k other than one, there are infinitely many n with two to the n congruent to k modulo n.
Estimates the least N forcing a monochromatic solution of a plus b equals c in every k-coloring of one through N, and asks whether it is exponential in k; open between c times 3.28 to the k and (e minus 1/6) times k factorial.
Asks whether, for all pairs of real numbers, the product of n with the distances from n times each number to the nearest integer has limit inferior zero.
Asks for the largest number of triples on n vertices with no four vertices carrying all four of their triples, Turán's tetrahedron problem; the density lies between Turán's 5/9 and a flag-algebra bound of 0.5615.
Determines the largest size of a set of points in d-dimensional space in which every three points form an isosceles triangle.
Estimates the smallest area such that every set of n points in the unit disk contains three points forming a triangle of at most that area.
Determines the fewest colors needed to color the plane so that no two points at distance one share a color.
Asks whether the set where a monic nonconstant complex polynomial has modulus at most one can be covered by circles whose radii sum to at most two.
Asks whether every set of N positive integers admits an angle where the sum of the cosines of its members times that angle is below a negative constant times root N; the site's wording over all integers fails at sets containing zero.
Determines the largest limit inferior of the ratio of the maximal power series term of a transcendental entire function to its maximum modulus on radius r.
Asks whether an entire power series whose exponents grow faster than linearly in the index must take every complex value infinitely often.
The value of the limiting growth rate per step of the number of self-avoiding walks of n steps from the origin in the k-dimensional integer lattice.
Asks whether the expected end-to-end distance of an n-step self-avoiding walk is of larger order than the square root of n in the plane, and at most of that order in every dimension at least three.
The order of the largest Sidon subset, one with no non-trivial equal pairwise sums, guaranteed inside every set of N real numbers.
Estimates the least N such that every two-coloring of the integers up to N contains a set of k numbers all of whose non-empty subset sums have the same color.
Estimates the largest subset of the integers up to N containing no r elements whose pairwise greatest common divisors are all equal, for r at least three.
The largest subset of the integers up to N containing no three distinct elements whose three pairwise least common multiples are all equal.
Estimates the least possible size of the set of ratios of each element to the greatest common divisor of a pair, over all sets of n natural numbers.
Asks whether the gap between consecutive Ramsey numbers of a triangle against a complete graph tends to infinity, and whether it is smaller than order k; open, with the gap known only to lie between 3 and k+1.
Asks whether, for all large m, the graph with m edges that is as complete as possible has the largest Ramsey number among graphs with m edges and no isolated vertices; open, while the site's wording, for every m, fails at two edges.