Representation of Primes in Arithmetic Progression by Binary Quadratic Forms
نویسندگان
چکیده
منابع مشابه
Rational Representations of Primes by Binary Quadratic Forms
Let q be a positive squarefree integer. A prime p is said to be q-admissible if the equation p = u2 + qv2 has rational solutions u, v. Equivalently, p is q-admissible if there is a positive integer k such that pk2 ∈ N , where N is the set of norms of algebraic integers in Q( √ −q). Let k(q) denote the smallest positive integer k such that pk2 ∈ N for all q-admissible primes p. It is shown that ...
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is a (nonconstant) arithmetic progression of positive integers. We consider a general binary quadratic form ax2 + bxy + cy' ( a , b , c E Z ) and ask the question "Can the form ax' + hxy + ry' represen1 every inleger in 1he arithmetic progression kNo + 1 for any natural numbers k and l?" In a sampling of books containing a discussion of binary quadratic forms [2]-[9], we did not find this qustl...
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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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In this article, we consider the representation of prime powers by binary quadratic forms of discriminant D = −2q1 . . . qt where the product of primes q1 . . . qt ≡ 3 (mod 4), for instance if it is of special RichaudDegert type n2 ± 2 for odd n’s, n2 − 1 for even n’s. We consider all the ambiguous classes and Q( √|D0|), where D0 is the fundamental discriminant and we obtain a general criterion...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 1993
ISSN: 0022-314X
DOI: 10.1006/jnth.1993.1065