On primes in arithmetic progressions
نویسندگان
چکیده
Let d > 4 and c ∈ (−d, d) be relatively prime integers, and let r(d) be the product of all distinct prime divisors of d. We show that for any sufficiently large integer n (in particular n > 24310 suffices for 4 6 d 6 36) the least positive integer m with 2r(d)k(dk− c) (k = 1, . . . , n) pairwise distinct modulo m is just the first prime p ≡ c (mod d) with p > (2dn − c)/(d − 1). We also conjecture that for any integer n > 4 the least positive integer m such that |{k(k − 1)/2 mod m : k = 1, . . . , n}| = |{k(k − 1)/2 mod m + 2 : k = 1, . . . , n}| = n is just the least prime p > 2n− 1 with p+ 2 also prime.
منابع مشابه
Arithmetic Progressions of Primes in Short Intervals
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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