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By P. D. T. A. (-co, M. B. Barban, U. V. Linnik and N. G. Chudakov


On primes and powers of a fixed integer. J. London Math. Soc. (2) 67 (2003), no. 2, 365–379. Let b be a power of an odd prime. Then it is shown that there are infinitely many triples (p, q, k), with k a positive integer and p, q primes, such that b k p + 1 q + 1, b2k < q < b 55k. This can be seen as an approximation to the conjecture that there should be infinitely many representations b = (p + 1)/(q + 1). The idea behind the proof lies in considering those primes p for which bk | p + 1, and to show that this happens sufficiently frequently that some value of b−k (p + 1) must be repeated. The most important tool in the proof is a variant of a result of M. B. Barban, Yu. V. Linnik and N. G. Chudakov [Acta Arith. 9 (1964), 375–390; MR0171766 (30 #1993)]. This gives a strong form of the prime number theorem in arithmetic progressions, when the modulus is nearly a prime power

Topics: 6. W. J. Ellison and M. Mend‘es-France, Les nombres premiers, Publications de l’Institut de
Year: 2010
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