等差素数列について

概要

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A finite ncreasing sequence {pn}, (n=1, 2, 3, ・・・, t) of prime numbers, (t〓3), is called an arithmetic progression of primes with a common differnce d, denoted by Ap(p_1) or Ap(p_1, d), if p_<k+1>-p_k=d∀k, 1〓k〓t-1 and p_t+d is a composite number. Then t is called the size of Ap(p_1). Ap(2) does not exist. For any Ap(p_1, d), prime p>2, its size is at most p, and d is an even number different from multiples of p. In this note we report various data on arithmetic progressions of primes gathered from long computation using personal computers. For example the following Ap(p, d) given the pairs (p, d) are of size 14, which are the largest size among primes p<10^7 and common differnces d<10^8. [table] We propose the following Conjecture For any prime p>2 we have 1) There exist infinitely many Ap(P). 2) There exist an Ap(P) of size p. The above 2) is true for p=3, 5, 7, 11.
2007-03-31