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Explicit bounds for primes in residue classes
 Math. Comp
, 1996
"... Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree1 prime p of K su ..."
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Cited by 16 (1 self)
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Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree1 prime p of K such that p = σ, satis
Notes on Dickson’s Conjecture
, 2009
"... In 1904, Dickson [5] stated a very important conjecture. Now people call it Dickson’s conjecture. In 1958, Schinzel and Sierpinski [14] generalized Dickson’s conjecture to the higher order integral polynomial case. However, they did not generalize Dickson’s conjecture to the multivariable case. In 2 ..."
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In 1904, Dickson [5] stated a very important conjecture. Now people call it Dickson’s conjecture. In 1958, Schinzel and Sierpinski [14] generalized Dickson’s conjecture to the higher order integral polynomial case. However, they did not generalize Dickson’s conjecture to the multivariable case. In 2006, Green and Tao [13] considered Dickson’s conjecture in the multivariable case and gave directly a generalized HardyLittlewood estimation. But, the precise Dickson’s conjecture in the multivariable case does not seem to have been formulated. In this paper, based on the idea in [15], we will try to complement this and give an equivalent form of Dickson’s Conjecture, furthermore, generalize it to the multivariable case or a system of affinelinear forms on N k. We also give some remarks and evidences on conjectures in [15].
On the Infinitude of Some Special Kinds of Primes
, 2009
"... The aim of this paper is to try to establish a generic model for the problem that several multivariable numbertheoretic functions represent simultaneously primes for infinitely many integral points. More concretely, we introduced briefly the research background–the history and current situation–f ..."
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The aim of this paper is to try to establish a generic model for the problem that several multivariable numbertheoretic functions represent simultaneously primes for infinitely many integral points. More concretely, we introduced briefly the research background–the history and current situation–from Euclid’s second theorem to GreenTao theorem. We analyzed some equivalent necessary conditions that irreducible univariable polynomials with integral coefficients represent infinitely many primes, found new necessary conditions which perhaps imply that there are only finitely many Fermat primes, generalized Euler’s function, the primecounting function and SchinzelSierpinski’s Conjecture and so on, obtained an analogy of the Chinese Remainder Theorem. By proposed obtrusively several conjectures, we gave a new way for determining the existence of some special kinds of primes. Finally, we proposed sufficient and necessary conditions that several multivariable numbertheoretic functions represent simultaneously primes for infinitely many integral points. Nevertheless, this is only a beginning and it miles to go. We hope that number theorists consider further it.