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.
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 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.
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 a factorial can equal a product of two or more smaller factorials, each at least two, only finitely often.
Asks whether any k consecutive composite integers can always be assigned distinct primes, one dividing each of them.
Asks whether infinitely many central binomial coefficients are coprime to one hundred and five.
Asks whether the sum of the reciprocals of the primes up to n that do not divide the central binomial coefficient of n is bounded by a constant.
Asks how long an interval of integers can be when the largest prime dividing its product appears at least twice, and whether that length can be arbitrarily large.
Asks whether, for every k, infinitely many primes p are the largest prime factor of the product of p squared through p squared plus k.
Asks whether the largest value of a composite number below n plus its least prime factor exceeds n for all large n, and whether the excess grows without bound.
Asks whether a binomial coefficient, other than the trivial ones, can be a product of consecutive primes infinitely often.
Asks whether two disjoint blocks of more than three consecutive integers can have equal products only finitely often, and whether such cases can be classified.
Asks whether every positive integer n admits some k for which the product of the first k integers from n divides the product of the next k.
Determines the behavior of the smallest spread between largest and smallest factor when a factorial is written as a product of distinct increasing integers.
Asks whether, for every k, some n makes the product of the k plus one integers from n minus k to n divide the central binomial coefficient of n.
Asks for the average and typical size of the largest excess of a sum of numbers whose factorials divide n factorial over n, for each fixed count k of terms.
Asks for which integers a and primes p the power of p dividing a sum of increasing factorials starting at a is bounded, and how that bound behaves.
Asks whether only finitely many powers of two are written with just the digits zero and one in base three.
Asks whether the number of totient iterations needed to reach one, divided by the logarithm of n, has a distribution or is almost always constant.
Asks how many iterations of the map sending n to Euler's totient of n plus one are needed to reach a prime, and how often a given prime is reached.
Asks whether the k-th iterate of the sum-of-divisors function, taken to the power one over k, tends to infinity for every integer n at least two.