Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
606 of 1,221 problems match
Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
Whether the number of consecutive pairs of an infinite set with least common multiple at most x is O(x^{1/2}), and how large the liminf of that count over x^{1/2} can be; both answered by Erdős and Szemerédi in 1980.
The largest subset of one through N with all pairwise least common multiples at most N, asymptotically the square root of 9N/8 by Chen, and whether Erdős's construction attains it, which Chen and Dai refute infinitely often.
Whether a set whose reciprocal sum grows faster than log log x forces the normalized sum of reciprocals of pairwise least common multiples to blow up; disproved by Tao, whose construction also gives the optimal growth threshold.
How many values are shared by the two sets of products k times m minus k and l times n minus l for m different from n, and whether this count can be large or must be very small.
Asks whether, for every k, an infinite set of integers has some integer below x with more divisors from the set than any power of the set's reciprocal sum.
The growth rate of the density of integers having a divisor strictly between n and twice n.
The largest family of subsets of one through n in which no set is the union of two other distinct members of the family.
Asks whether, for almost all n, the number of dyadic ranges containing a divisor of n is an arbitrarily small fraction of the total number of divisors.
Asks whether, for almost all n, the number of pairs of divisors within a factor of two of each other is an arbitrarily small fraction of the divisor count.
Asks whether, for all large n, some symmetric pair of primes around the nth prime has product exceeding the square of the nth prime.
Whether some epsilon > 0 gives infinitely many n with every prime up to (2 + epsilon) log n dividing the product of the next log n integers: yes, by a 2026 AI construction the site accepted; Erdős's opposite 1979 form: no.
Estimates the largest v such that no integer strictly between u and v is built only from primes dividing the product of u and v.
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.
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).
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.
Asks whether the reciprocal sum converges over integers that are sums of distinct proper divisors of themselves while no proper divisor has that property.
Asks whether some finite starting set of primes grows without bound under repeated adjunction of all primes that are sums of three distinct members; yes, by Vinogradov's three-primes theorem, an observation the site accepted.
Asks whether the positive integers can be permuted so that every two consecutive terms sum to a prime; yes, by an unpublished construction of Odlyzko reported by Erdős and Graham in 1980 and accepted by the site.
Which set theoretic assumptions allow a three-coloring of the plane in which every uncountable set contains a pair of points of each color.
Asks whether the set of sums of distinct pairs from a subset of the integers modulo a prime has size at least twice the subset size minus three, or the prime (the Erdős–Heilbronn conjecture); proved in 1994.
Asks whether the image of an integer polynomial of degree at least two admits a unique additive complement in the integers.
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....
Asks whether iterating a family of affine maps from the value one must repeat an element when the reciprocals of the multipliers sum to over one; yes, by Klarner's 1982 theorem, its 2022 extension and a 2025 thread proof.
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.
Asks whether every k-coloring of the first N integers leaves a positive proportion of them expressible as a sum of two distinct integers of one color.