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.