Euler’s function on products of primes in a fixed 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...
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The following is a quick set of notes of some properties of Dirichlet characters, in particular, how they are used to prove the infinitude of primes in arithmetic progressions. These notes are from from An Invitation to Modern Number Theory, by myself and Ramin Takloo-Bighash. As this is a modified snippet from the book, references to other parts of the book are displayed as ??. 1. Dirichlet Ch...
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It has been a long conjecture that there are arbitrarily long arithmetic progressions of primes. As of now, the longest known progression of primes is of length 26 and was discovered by Benoat Perichon and PrimeGrid in April, 2010 ([1]): 43142746595714191+23681770·223092870n for n = 0, 1, · · · , 25. Many mathematicians have spent years trying to prove (or disprove) this conjecture, and even mo...
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In 1967 the first set of 6 consecutive primes in arithmetic progression was found. In 1995 the first set of 7 consecutive primes in arithmetic progression was found. Between November, 1997 and March, 1998, we succeeded in finding sets of 8, 9 and 10 consecutive primes in arithmetic progression. This was made possible because of the increase in computer capability and availability, and the abili...
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It is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression. In 1967, the first such sequence of 6 consecutive primes in arithmetic progression was found. Searching for 7 consecutive primes in arithmetic progression is difficult because it is necessary that a prescribed set of at least 1254 numbers between the first and last prime all be composi...
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 2019
ISSN: 0025-5718,1088-6842
DOI: 10.1090/mcom/3463