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.
Asks whether the fewest distinct unit fractions with denominators at least N summing to one exceeds e minus one times N by an amount tending to infinity.
Estimates the largest subset of one through N in which no reciprocal is a sum of reciprocals of other distinct members, and asks whether it is about half of N.
Estimates the largest subset of one through N with no distinct members where one reciprocal is the sum of two others, and asks whether it is about half of N.
Asks whether two finite sets of primes exist whose sums of reciprocals multiply together to give one.
Asks whether the least non-zero distance from one to a subset sum of reciprocals of one through N decays like e to the power of minus a constant times N.
Asks whether a large multiset of integers whose reciprocals sum above K always has a subset whose reciprocals sum to at most one but within e to the minus cK.
Asks whether infinitely many sums of reciprocals of distinct primes equal one minus the reciprocal of an integer.
Asks whether signs of minus one, zero or one can always make the signed sum of reciprocals up to n non-zero yet smaller than a constant over two to the n.
The size of the largest subset of one through N carrying signs whose signed reciprocals sum to zero while no proper non-empty subset sums to zero.
Estimates the number of ways an integer is a sum of k many kth powers, and asks whether it exceeds n to a fixed positive power infinitely often.
Asks whether the integers up to x that are sums of k kth powers number at least x to the power 1 minus epsilon, and whether sums of m such powers, m below k, number at least a constant times x to the m over k.
Asks whether some polynomial with integer coefficients has all sums of two of its values at distinct nonnegative integers distinct.
Asks whether the number of integers up to x that are sums of three nonnegative kth powers is at least a constant times x to the power three over k.
The largest possible value of the limiting ratio of the counting function of an infinite Sidon set to the square root of N.
Determines which conditions on a set of integers force the differences that occur infinitely often to have bounded gaps.
The best function f for which every n is a sum of two integers having no prime factor larger than f of n; Erdős asked whether n^epsilon suffices, still open, and whether even n^(1/3) does, which Balog's bound answers yes.
Characterizes the pairs of positive-density sets of integers whose sumset has density exactly the sum of their densities.
Characterizes when a basis has a restricted order, using distinct summands only, and whether that order is bounded in terms of the ordinary order.
The growth rate of the greedy Sidon sequence, and whether its counting function is at least N to the power one half minus any epsilon.
Asks what can be said about the sequence whose terms are the least integers uniquely expressible as a sum of two earlier terms, its density and its gaps.
Asks whether infinitely many k make the completeness threshold of the kth powers larger than that of the next powers.
Determines for which positive t and alpha the integer parts of t times alpha to the n form a complete sequence, distinct terms summing to all large integers.
Asks whether some positive constant makes every planar measurable set of at least that measure contain the vertices of a triangle of area one.
Asks whether the rounded-down doubling multiples of two reals with irrational ratio form a complete sequence, and the same for a base between one and two; the first question is answered yes, the second has two readings.
The growth rate of the largest number of integers up to n whose sums over blocks of consecutive terms are all distinct, and whether it is smaller than n.