From tonyforbes@ltkz.demon.co.ukTue Jan 16 22:50:46 1996 Date: Tue, 16 Jan 1996 15:34:06 EST From: Tony Forbes To: Multiple recipients of list NMBRTHRY Subject: A small collection of prime k-tuplets A small collection of prime k-tuplets A prime k-tuplet is a sequence of consecutive primes p(1), p(2), ..., p(k) such that s(k) = p(k) - p(1) is minimal in the sense that a smaller value of s(k) is ruled out by divisibility considerations. The minimal values of s(k) are: s(2)=2 (twins), s(3)=6, s(4)=8, s(5)=12, s(6)=16, s(7)=20, s(8)=26, s(9)=30, s(10)=32, s(11)=36, s(12)=42, s(13)=48, s(14)=50, and so on. I report here an assortment of prime k-tuplets (k>2) found by me over the past couple of years. All were (or will be) published in "M500" the newsletter of the M500 Society, a mathematics society for students, staff and friends of the Open University of Great Britain. Computer hardware: 486 DX/33MHz, later, 486 DX4/100MHz. Software: Yuji Kida's UBASIC, supplemented by programs written in PC Assembler language for (i) k-tuple seiving by primes up to about a million, and (ii) speeding up the Fermat test for numbers of shape 2^n + m, m < 2^64. The notation N + 0, B2, ..., Bk, means that the prime k-tuplet consists of N, N + B2, ..., N + Bk. "prp" means probable primes that have passed the Fermat test to several bases. All others have been verified with UBASIC's version of the Adleman-Pomerance-Rumely-Cohen-Lenstra test. 3-tuplets 2^3456 + 5661177712051 + 0, 2, 6 (1995, 1041 digits, M500 145) prp 2^2400 + 14906370057 + 0, 4, 6 (1994, 723 digits, M500 139) prp 2^2080 + 17304485518 + 0, 4, 6 (1994, 627 digits, M500 139) prp 2^2080 + 9595427487 + 0, 4, 6 (1994, 627 digits, M500 139) prp 4-tuplets 2^1056 + 1301655396715 + 0, 2, 6, 8 (1994, 318 digits, M500 140) 2^800 + 10169432335 + 0, 2, 6, 8 (1994, 241 digits, M500 140) prp 2^672 + 610470566785 + 0, 2, 6, 8 (1995, 203 digits, M500 146) prp 3 * 10^200 + 1984539701 + 0, 2, 6, 8 (1994, M500 137) 5-tuplet 2^576 + 79801763715655 + 0, 2, 6, 8, 12 (1995, 174 digits, M500 147) 6-tuplet 2 * 10^132 + 75543532187 + 0, 4, 6, 10, 12, 16 (1994, M500 137) 8-tuplets 15 * 10^52 + 11527947572567 + 0, 2, 6, 12, 14, 20, 24, 26 (1995, M500 145) 9 * 10^50 + 3219584035187 + 0, 2, 6, 12, 14, 20, 24, 26 (1994, M500 137) 28 * 10^37 + 115808692429393 + 0, 6, 8, 14, 18, 20, 24, 26 (1995, M500 146) 6 * 10^37 + 3191144505101 + 0, 2, 6, 8, 12, 18, 20, 26 (1995, M500 145) 6 * 10^37 + 1178750105771 + 0, 2, 6, 8, 12, 18, 20, 26 (1995, M500 145) 9-tuplet 22 * 10^35 + 111245519078161 + 0, 2, 6, 8, 12, 18, 20, 26, 30 (1995, M500 146) 10-tuplets 26 * 10^32 + 30697502738421 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (1995, M500 146) 12 * 10^28 + 14884968491321 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (1994, M500 137) 25 * 10^23 + 4104936514466137 + 0, 2, 6, 12, 14, 20, 24, 26, 30, 32 (1995, M500 146) 2^80 + 1051069612640371 + 0, 2, 6, 12, 14, 20, 24, 26, 30, 32 (1995, M500 146) 11-tuplets 300004226699607434083 + 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36 (1995, M500 146) 300001698148447235173 + 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36 (1995, M500 146) 300001141243282082683 + 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36 (1995, M500 146) 12-tuplet 300006711401831244037 + 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 (1995, M500 146) 13-tuplets 2725574911051687529 + 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 (1996, M500 t.a.) 2365201889521345991 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 (1995, M500 t.a.) 1599703625518603439 + 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 (1995, M500 t.a.) 1261453132631116259 + 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 (1995, M500 t.a.) I would dearly like to enlist someone's help to verify the titanic probable triplets 2^3456+5661177712051+0,2,6. (UBASIC's test only goes up to 844 digits.) Perhaps there is suitable software for the PC? A larger 4-tuplet beginning 10^400+34993836001 has been found by Warut Roonguthai (Number Theory List, September 1995). 14-tuplets beginning 21817283854511261 and 79287805466244209 have been found by Dimitrios Betsis & Sten Safholm (1982; see R. K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, 1994). -- Tony Forbes