Largest known twin primes and Sophie Germain primes

The numbers 242206083 · 238880 ± 1 are twin primes. The number p = 2375063906985 · 219380 − 1 is a Sophie Germain prime, i.e. p and 2p + 1 are both primes. For p = 4610194180515 · 25056 − 1, the numbers p, p + 2 and 2p + 1 are all primes. In the first days of October, 1995, Harvey Dubner [4] found the largest known twin primes with 5129 decimal digits. (Our earlier twin prime record was 697053813· 216352±1, with 4932 decimal digits; see [5]). Some days before he found the largest known Sophie Germain prime, with 5089 decimal digits (see [8], p. 330), beating our records 157324389 · 2 − 1 and 470943129 · 2 − 1 with 4931 and 4932 decimal digits, respectively, found in February 1995 (first appearing here in print). In the same days our twin prime program was already running to find large twins in the range with approximately 11700 decimal digits. Because the total expected running time was 1500–2000 CPU days on Sunr workstations with SuperSPARCr processors of Texas Instruments with clock frequency from 33 MHz to 60 MHz, we decided to stop the search for some time and to use our new combined test program which is able to search for twin primes and Sophie Germain primes simultaneously, with half the running time. The idea of combining the two tests originated from Dubner [4]. Since our estimate for the running time of this new test to find twin primes and Sophie Germain primes with ≈ 5850 decimal digits was somewhat below 100 CPU days (total), we thought it would be more realistic to do this search first and later to continue with the combined test in the range with ≈ 11700 decimal digits. To test the combined sieve program and to do the new, combined sieving for twin and Sophie Germain primes needed some days, and until this we let the “old” program run. In the last day before stopping the old program we found the twin primes 242206083 · 2 ± 1 (11713 digits). After this we stopped the old program, and started to run the new combined test, which found the largest known Sophie Germain prime p = 2375063906985 · 2 − 1 (5847 digits). Large Sophie Germain primes p congruent to 3 mod 4 lead to large composite Mersenne numbers. So 2p−1 for p = 2375063906985·219380−1 is the largest known composite Mersenne number. In an e-mail message Harvey Dubner suggested looking for primes p for which p + 2 and 2p + 1 are primes, too. Our new combined test program was able to do this without any changes, so we also have done a search in this direction and found Received by the editor April 7, 1997 and, in revised form, February 5, 1998. 1991 Mathematics Subject Classification. Primary 11-04; Secondary: 11A41. rSun is a registered trademark of Sun Microsystems Inc. rSPARC is a registered trademark of SPARC International. c ©1999 American Mathematical Society