Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
1,221 problems
Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
Asks whether the numbers p of n plus one over n, for a rational polynomial p with positive leading coefficient, stay complete after any finite set is removed.
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 a planar measurable set of infinite measure must contain the vertices of an isosceles trapezoid of area one, or of other prescribed shapes.
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.
Asks whether some sequence growing at least geometrically has finite sums of reciprocals of its terms covering every rational in some open interval.
Asks whether some positive c gives, for all large n, integers up to n whose sums over blocks of consecutive terms take at least c times n squared values.
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.
Asks whether some infinite sequence of integers writes every large n as a sum of consecutive terms at least twice, or in ways tending to infinity.
The density of the sequence starting at n in which each later term is the least integer that is not a sum of consecutive earlier terms.
The growth rate of the fewest classes needed to partition the numbers below n so that n is never a sum of distinct members of one class.
The largest subset of the integers up to c times n that has no subset summing to n, and whether its size varies irregularly with n.
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.
Asks whether only finitely many families of disjoint integer intervals, each of length at least four, have the product of all their members equal to a square.
Asks whether three consecutive positive integers can all be powerful, meaning every prime dividing such a number divides it at least twice.
Asks whether one of any two consecutive powerful numbers must be a square, and whether such pairs up to x number at most a power of the logarithm of x.
Asks whether some integer divisible by the square of each of its prime factors is followed by one divisible by the cube of each; Erdős and Graham's companion question, with the cube first, is answered by 12167, 12168.
Asks whether the product of the powerful parts of k consecutive integers near n is at most about n squared, for every fixed k.
Asks how large the largest prime factor of the product of n and n plus one is.
Asks whether every large n has k consecutive n^epsilon-smooth integers up to n; trivially true as worded, it is proved in both nontrivial readings, each member smooth to its own power or the run inside [n/2, n].
Asks whether infinitely many integers n have both n and n plus one with largest prime factor below their own square root.
Asks whether the integers n whose largest prime factor is smaller than that of n plus one have density one half.
Asks whether infinitely many integers n have the largest prime factors of n, n plus one, and n plus two in strictly decreasing order.
Asks whether a factorial can equal a product of two or more smaller factorials, each at least two, only finitely often.
Determines how many integers up to n need exactly k factorials, with the largest being that integer's, to form a square product, for k from three to six.
Asks whether any k consecutive composite integers can always be assigned distinct primes, one dividing each of them.