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ON PRIMES IN QUADRATIC PROGRESSIONS
, 2007
"... Abstract. We verify the HardyLittlewood conjecture on primes in quadratic progressions on average. The ..."
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Abstract. We verify the HardyLittlewood conjecture on primes in quadratic progressions on average. The
Sieve Methods
"... Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after th ..."
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Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after the pioneering work of Viggo Brun (see [Bru16],[Bru19], [Bru22]). Using his formulation of the sieve Brun proved, that the sum
ON THE GREATEST AND LEAST PRIME FACTORS OF n!+l
"... Much work has been done on obtaining estimates from below for the greatest prime factors of the terms of certain sequences of integers. Let P(W) denote the greatest prime factor of nz and letf(x) be any irreducible polynomial with degree> 1 and integer coefficients. It can easily be deduced from Sie ..."
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Much work has been done on obtaining estimates from below for the greatest prime factors of the terms of certain sequences of integers. Let P(W) denote the greatest prime factor of nz and letf(x) be any irreducible polynomial with degree> 1 and integer coefficients. It can easily be deduced from Siegel’s work [ & 121 that P(f (x)) + w as x 3 co and recently Sprindzhuk and Kotov [7], using deep techniques of Baker, have shown that indeed P(f (x))> c log log x for all integers x where L ’ = c(f)> 0; the case of quadratic and cubic f was in fact covered by earlier works of Schinzel [lo] and Keates [6]. In another context Birkhoff and Vandiver Cl] proved, by elementary methods, that for distinct positive integers a, b, P(a”b”)> nf 1 for all integers 12> 6. Recently, again using techniques of Baker, the second author [14] obtained some new results in this connexion; for instance, for the Fermat numbers we have (see [15]) P(Y”$
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"... On the greatest prime factor of integers of the form ab + 1 by C.L. Stewart ∗ to Professor András Sárközy on the occasion of his sixtieth birthday Abstract: Let N be a positive integer and let A and B be dense subsets of {1,..., N}. The purpose of this paper is to establish a good lower bound for th ..."
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On the greatest prime factor of integers of the form ab + 1 by C.L. Stewart ∗ to Professor András Sárközy on the occasion of his sixtieth birthday Abstract: Let N be a positive integer and let A and B be dense subsets of {1,..., N}. The purpose of this paper is to establish a good lower bound for the greatest prime factor of ab + 1 as a and b run over the elements of A and B respectively.
The Largest Prime Factor of X 3 + 2
"... It is conjectured that if f(X) is an irreducible integer polynomial with positive leading coefficient, then f(n) takes infinitely many prime values, providing only that f(n) has no fixed prime divisor. As an approximation to this conjecture one might ask whether f(n) has a very large prime factor in ..."
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It is conjectured that if f(X) is an irreducible integer polynomial with positive leading coefficient, then f(n) takes infinitely many prime values, providing only that f(n) has no fixed prime divisor. As an approximation to this conjecture one might ask whether f(n) has a very large prime factor in infinitely many cases.
ON PRIMES REPRESENTED BY QUADRATIC POLYNOMIALS
, 2008
"... Abstract. This is a survey article on the HardyLittlewood conjecture about primes in quadratic progressions. We recount the history and quote some results approximating this hitherto unresolved conjecture. ..."
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Abstract. This is a survey article on the HardyLittlewood conjecture about primes in quadratic progressions. We recount the history and quote some results approximating this hitherto unresolved conjecture.