Sums of primes and squares of primes in short intervals
نویسندگان
چکیده
منابع مشابه
Sums of Primes and Squares of Primes in Short Intervals
Let H2 denote the set of even integers n 6≡ 1 (mod 3). We prove that when H ≥ X, almost all integers n ∈ H2 ∩ (X,X + H] can be represented as the sum of a prime and the square of a prime. We also prove a similar result for sums of three squares of primes.
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Contrary to what would be predicted on the basis of Cramér’s model concerning the distribution of prime numbers, we develop evidence that the distribution of ψ(x + H) − ψ(x), for 0 ≤ x ≤ N , is approximately normal with mean ∼ H and variance ∼ H logN/H , when N ≤ H ≤ N1−δ .
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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.
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Green and Tao proved that the primes contains arbitrarily long arithmetic progressions. We show that, essentially the same proof leads to the following result: If N is sufficiently large and M is not too small compared with N , then the primes in the interval [N, N + M ] contains many arithmetic progressions of length k.
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In this paper we show that sieve methods used previously to investigate primes in short intervals and corresponding Goldbach type problems can be modified to obtain results on primes in Beatty sequences in short intervals.
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ژورنال
عنوان ژورنال: Monatshefte für Mathematik
سال: 2008
ISSN: 0026-9255,1436-5081
DOI: 10.1007/s00605-008-0047-1