منابع مشابه
Bubbles of Congruent Primes
In [15], Shiu proved that if a and q are arbitrary coprime integers, then there exist arbitrarily long strings of consecutive primes which are all congruent to a modulo q. We generalize Shiu’s theorem to imaginary quadratic fields, where we prove the existence of “bubbles” containing arbitrarily many primes which are all, up to units, congruent to a modulo q.
متن کاملStrings of congruent primes in short intervals
Fix > 0, and let p1 = 2, p2 = 3, . . . be the sequence of all primes. We prove that if (q, a) = 1, then there are infinitely many pairs pr, pr+1 such that pr ≡ pr+1 ≡ a mod q and pr+1 − pr < log pr. The proof combines the ideas of Shiu and of Goldston–Pintz–Yıldırım.
متن کاملOn Congruent Primes and Class Numbers of Imaginary Quadratic Fields
We consider the problem of determining whether a given prime p is a congruent number. We present an easily computed criterion that allows us to conclude that certain primes for which congruency was previously undecided, are in fact not congruent. As a result, we get additional information on the possible sizes of Tate-Shafarevich groups of the associated elliptic curves. We also present a relat...
متن کاملOptimal (9v, 4, 1) optical orthogonal codes with v a product of primes congruent to 1 modulo 4
Optimal (9p, 4, 1) optical orthogonal codes are constructed for all primes p congruent to 1 modulo 4. Direct constructions with explicit codewords are presented for the case p ≡ 13 mod 24, and Weil’s theorem on character sums is used to settle the cases p ≡ 1, 5, 17 mod 24. By applying a known recursive construction, optimal (9v, 4, 1) and (45v, 4, 1) optical orthogonal codes are obtained for a...
متن کاملProportion Congruent effects in the absence of Sequential Congruent effects
A debated question in the cognitive control field is whether cognitive control is best conceptualized as a collection of distinct control mechanisms or a single general purpose mechanism. In an attempt to answer this question, previous studies have dissociated two well-known effects related to cognitive control: sequential congruence and proportion congruent effects. In the present experiment, ...
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ژورنال
عنوان ژورنال: Mathematical Proceedings of the Cambridge Philosophical Society
سال: 2014
ISSN: 0305-0041,1469-8064
DOI: 10.1017/s0305004114000425