From 70372.1170@compuserve.comWed Aug 30 10:22:26 1995 Date: Tue, 29 Aug 1995 21:44:24 EDT From: Harvey Dubner <70372.1170@compuserve.com> To: Multiple recipients of list NMBRTHRY Subject: 7 consec primes in AP Harvey Dubner and Harry Nelson would like to announce that we have found 7 consecutive primes in arithmetic progression. The previous record was 6; the first such sequence was found by Lander and Parkin (Consecutive Primes in Arithmetic Progression, MathComp, 21, p.489, 1967) and consisted of 9-digit numbers. Sol Weintraub summarized the status in the Journal of Recreational Math. Vol 25(3), 1993. For 7 primes in arithmetic progression it is easy to show that the smallest possible common difference is 210. The problem really separates into two sub-problems: 1. Find 7 primes in AP with common difference = 210. 2. Find 1254 numbers between the first and last prime that are all composite, except of course for the primes in AP. (1) suggests looking for small numbers, while (2) suggests searching for large numbers. Based on straight probability, searching should take place in the 50-digit range and would take about 10 computer years on a 486/66. However, Harry Nelson handled a problem similar to (2) in JRM, Vol. 8(1), 1975 in which he developed a system of simultaneous modular equations to guarantee that a sequence of numbers will all be composite. We solved such a system of 48 equations to determine optimum search starting points in the 90-digit range. The search took about 52 computer-days which was quite close to the estimated time including appropriate sieving. Up to 7 computers were used over a two week period. The computers included Cruncher accelerator boards which were only marginally helpful in this 90-100 digit range. We are preparing a paper which we hope will be available shortly. The 7 consecutive primes in AP are found by computing m, the product of primes up to the 48th prime, m = 36700973182733191646503456555013673233980031295533178261946245703998_ 8073311157667212930 Next, find x, the solution for the 48 modular equations, x = 11893061343242550473160091662536053989417322887015941546297601405680_ 9082107460202605690 Then, after appropriate sieving, search over all N such that the first prime is, P1 = x + N*m + 1. When 7 primes with AP = 210 are found, the in-between numbers are all tested for compositeness. If all are composite the search is successful. Success occurred at N=2968677222 P1 is the 97 digit number, P1 = 1089533431247059310875780378922957732908036492993138195385213105561_ 742150447308967213141717486151 P2 = P1 + 210, P3 = P2 + 210, ..... P7 = P6 + 210 Harvey Dubner PS: I cannot resist. If anyone wants more information about the Cruncher, just ask.