Gaps of Smallest Possible Order between Primes in an Arithmetic Progression
نویسندگان
چکیده
منابع مشابه
Primes in arithmetic progression
Prime numbers have fascinated people since ancient times. Since the last century, their study has acquired importance also on account of the crucial role played by them in cryptography and other related areas. One of the problems about primes which has intrigued mathematicians is whether it is possible to have long strings of primes with the successive primes differing by a fixed number, namely...
متن کاملGaps between Prime Numbers and Primes in Arithmetic Progressions
The equivalence of the two formulations is clear by the pigeon-hole principle. The first one is psychologically more spectacular: it emphasizes the fact that for the first time in history, one has proved an unconditional existence result for infinitely many primes p and q constrained by a binary condition q − p = h. Remarkably, this already extraordinary result was improved in spectacular fashi...
متن کاملSmall gaps between primes
The twin prime conjecture states that there are infinitely many pairs of distinct primes which differ by 2. Until recently this conjecture had seemed to be out of reach with current techniques. However, in 2013, the author proved that there are infinitely many pairs of distinct primes which differ by no more than B with B = 7 · 107. The value of B has been considerably improved by Polymath8 (a ...
متن کاملBounded gaps between primes
It is proved that lim inf n→∞ (pn+1 − pn) < 7× 10, where pn is the n-th prime. Our method is a refinement of the recent work of Goldston, Pintz and Yildirim on the small gaps between consecutive primes. A major ingredient of the proof is a stronger version of the Bombieri-Vinogradov theorem that is applicable when the moduli are free from large prime divisors only (see Theorem 2 below), but it ...
متن کاملLong Gaps between Primes
Let pn denotes the n-th prime. We prove that max p n+16X (pn+1 − pn) ≫ logX log logX log log log logX log log logX for sufficiently large X , improving upon recent bounds of the first three and fifth authors and of the fourth author. Our main new ingredient is a generalization of a hypergraph covering theorem of Pippenger and Spencer, proven using the Rödl nibble method. CONTENTS
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: International Mathematics Research Notices
سال: 2016
ISSN: 1073-7928,1687-0247
DOI: 10.1093/imrn/rnw013