Primes

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E0004/: Asks whether prime gaps exceed any given constant times log n times a slowly growing factor built from repeated logarithms infinitely often.

E0005/: Asks whether, for every constant C at least 0, the gap after the nth prime divided by log n tends to exactly C along some sequence of indices n.

E0006/: Asks whether there are infinitely many n for which three consecutive prime gaps are strictly increasing.

E0015/: Asks whether the alternating sum of n divided by the nth prime converges; open, with Tao's proof of convergence conditional on a strong Hardy-Littlewood prime tuples conjecture.

E0017/: Asks whether infinitely many primes p have every even number up to p minus 3 expressible as a difference of two primes not exceeding p.

E0049/: Asks whether a set of integers up to N on which Euler's totient function is strictly increasing has at most (1+o(1))π(N) elements, or even o(N); Erdős's exact conjecture, that the primes are a largest such set, is open.

E0218/: Asks whether consecutive prime gaps increase half the time and decrease half the time, and whether two consecutive gaps are equal infinitely often.

E0233/: Asks whether the sum of the squares of the first N prime gaps is at most a constant times N times the square of the logarithm of N.

E0234/: Asks whether the density of primes whose gap to the next prime is below c times the logarithm exists for every c and varies continuously in c.

E0236/: Asks whether the number of ways to write n as a prime plus a power of two is always small compared with the logarithm of n.

E0237/: Asks whether any set of integers with at least logarithmically many elements up to N gives some integers unboundedly many representations as prime plus member.

E0238/: Asks whether for any positive constants there are, below every large x, more than a multiple of log x consecutive primes that are pairwise far apart.

E0240/: Asks whether some infinite set of primes has the property that the gaps between consecutive integers built only from those primes tend to infinity.

E0244/: Asks whether, for a constant greater than 1, the integers formed as a prime plus the integer part of a power of that constant have positive density.

E0427/: Asks whether, for every n and d, some run of consecutive primes starting after the n-th prime has sum divisible by d.

E0428/: Asks for a set with positive relative density among the primes such that, for infinitely many n, subtracting each member from n always gives a prime.

E0431/: Asks whether two infinite sets have a sumset that agrees with the set of primes apart from finitely many exceptions.

E0454/: Asks whether the least sum of a symmetric pair of primes around the nth prime exceeds twice the nth prime by an unbounded amount infinitely often.

E0458/: Asks whether the least common multiple up to one below the next prime is always less than the previous prime times the least common multiple up to it.

E0459/: 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.

E0461/: Asks whether the number of distinct smooth parts, using primes below t, of the t consecutive integers after n is always at least a constant times t.

E0462/: Asks whether the sum of the least prime factor over n, over a short interval near x, is always bounded below by a constant for large x.

E0463/: Asks whether some function tending to infinity admits, for every large n, a composite number above n plus that function but below n plus its least prime factor.

E0680/: Asks whether every sufficiently large n admits some k for which the least prime factor of n plus k exceeds k squared plus one.

E0681/: Asks whether every large n has some k for which n plus k is composite and the least prime factor of n plus k exceeds k squared.

E0682/: Asks whether almost every n admits an integer strictly between the nth and next prime whose least prime factor is at least the gap between those primes.

E0779/: Asks whether, for the product P of the first n primes, there is always a prime p between the n-th prime and P such that P plus p is prime.

E0850/: Asks whether two distinct integers can agree in prime factors, with their successors also agreeing and the next integers after those agreeing too.

E0852/: Estimates the longest run of pairwise distinct consecutive prime gaps starting at an index below x, asking whether it exceeds a power of log x and whether it is o(log x).

E0853/: Asks whether the smallest even number missing from the first x prime gaps tends to infinity, and whether it grows faster than log x.

E0855/: Asks whether the number of primes up to x plus y is at most the number up to x plus the number up to y, for all large x and y.

E0860/: Estimates the shortest interval length that always contains distinct integers, one divisible by each prime up to n.

E0890/: Asks whether the summed count of large distinct prime factors over k consecutive integers is infinitely often at most k, and about its extreme growth rate.

E0950/: Asks for the limit inferior, limit superior and growth of the sum of the reciprocals of n minus p taken over all primes p below n.

E1055/: Concerns the classification of primes into classes by repeatedly factoring p plus one, starting from primes whose only such factors are two and three.

E1059/: Asks whether infinitely many primes p have p minus k factorial composite for every k with k factorial less than p.

E1137/: Asks whether the largest product of two consecutive prime gaps below x is negligible compared with the square of the largest prime gap below x.

E1138/: Asks whether an interval of length a constant times the maximal prime gap below x contains the expected number of primes, for y between x halved and x.

E1139/: Asks whether the gaps in the sequence of integers with at most two prime factors are infinitely often much larger than the logarithm of the index.

E1141/: Asks whether infinitely many n have n minus k squared prime for every k coprime to n with k squared below n.

E1142/: Asks whether infinitely many n, or any n above one hundred and five, make n minus every power of two greater than one and less than n prime.

E1143/: Estimates how many multiples of at least one of finitely many given primes every interval of k consecutive positive integers must contain.

E1184/: Asks whether the count of integers just above n whose largest prime factor exceeds k follows the prediction given by the Dickman function.

E1200/: Asks for primes below x with bounded reciprocal sum together with residues covering every integer less than x.

E1201/: Asks whether, for every epsilon and eta, some k makes the largest prime factor of n(n+1)...(n+k) exceed n to the one minus epsilon for a set of n of density at least 1 - eta, the density read as lower density following the site's curator.

E1202/: Asks whether some k exists so that sieving half the residue classes modulo each of k primes below n to the one minus epsilon leaves few integers up to n.

E1212/: Asks for a path to infinity through coprime pairs above 1 with a composite coordinate, steps changing one coordinate by one; Erdős first asked it without the composite condition, which Stewart quickly answered yes.

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Distribution of the primes, prime gaps, and prime values and patterns in arithmetic sequences.

Site tags routed here: number theory, primes.