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The Legendre symbol has been used to construct sequences with ideal cross-correlation, but it was never used in the arithmetic cross-correlation. In this paper, a new class of generalized Legendre sequences are described and analyzed with respect to their period, distributional, arithmetic cross-correlation and distinctness properties. This analysis gives a new approach to study the connection between the Legendre symbol and the arithmetic cross-correlation. In the end of this paper, possible application...

Goldbach conjecture.

H.A. Pogorzelski (1977)

Journal für die reine und angewandte Mathematik

Goldbach numbers in sparse sequences

Jörg Brüdern, Alberto Perelli (1998)

Annales de l'institut Fourier

We show that for almost all $n \in N$ , the inequality $| p_{1} + p_{2} - \exp ((\log n)^{γ}) | < 1$ has solutions with odd prime numbers $p_{1}$ and $p_{2}$ , provided $1 < γ < \frac{3}{2}$ . Moreover, we give a rather sharp bound for the exceptional set.This result provides almost-all results for Goldbach numbers in sequences rather thinner than the values taken by any polynomial.

Goldbach numbers represented by polynomials.

Alberto Perelli (1996)

Revista Matemática Iberoamericana

Let N be a large positive real number. It is well known that almost all even integers in the interval [N, 2N] are Goldbach numbers, i.e. a sum of two primes. The same result also holds for short intervals of the form [N, N+H], see Mikawa [4], Perelli-Pintz [7] and Kaczorowski-Perelli-Pintz [3] for the choice of admissible values of H and the size of the exceptional set in several problems in this direction.One may ask if similar results hold for thinner sequences of integers in [N, 2N], of cardinality...

Goldbach’s problem with primes in arithmetic progressions and in short intervals

Karin Halupczok (2013)

Journal de Théorie des Nombres de Bordeaux

Some mean value theorems in the style of Bombieri-Vinogradov’s theorem are discussed. They concern binary and ternary additive problems with primes in arithmetic progressions and short intervals. Nontrivial estimates for some of these mean values are given. As application inter alia, we show that for large odd $n \neg \equiv 1 (6)$ , Goldbach’s ternary problem $n = p_{1} + p_{2} + p_{3}$ is solvable with primes $p_{1}, p_{2}$ in short intervals $p_{i} \in [X_{i}, X_{i} + Y]$ with $X_{i}^{θ_{i}} = Y$ , $i = 1, 2$ , and $θ_{1}, θ_{2} \geq 0.933$ such that $(p_{1} + 2) (p_{2} + 2)$ has at most $9$ prime factors.

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