March 15, 1998, Nancy (France)

	Dear all,

this is the second ecmnet newsletter. The first appeared on February 13th,
and both are available on the ecmnet home page [1].

MS-DOS version. Conrad Curry, who wants to factor Mersenne numbers, managed 
to produce a MS-DOS binary of GMP-ECM using DJGPP. This should enable one
to get a Windows version too, I guess (I must admit I'm not a PC specialist).

Cunningham numbers. NFSNet factored 5,239+ (1st hole, c147=p52*p96) and
6,230+ (3rd hole, c132=p50*p82) on February 25, 5,247- on March 9 (1st hole, 
c147=p39*p41*p67, an ECM miss!), and 5,303- (c132=p57*p75) on March 12.
On March 14, MullFac completed the factorization of 3,1041L (c94), which 
decomposes into a p46 and a p49. On February 24, ECMNET found a p35 in 3,401- 
(5th hole), and on March 11, another p35 in 3,353+ (2nd hole). On Friday, 
March 13 (a lucky day), ECMNET found a p45 in 2,599- (2nd hole). Since last 
newsletter, Sam Wagstaff published a list of 10 "most wanted" Cunningham 
cofactors, and 24 "more wanted" numbers. From this list, two "most wanted" 
numbers (2,599- and 5,247-) are whence already factored, and one "more wanted"
number (3,353+). If you want to do a massive attack with ECM to one of the 
remaining numbers, please tell me, as some of them were already tested up
factors of 35 or 40 digits. Michel Quercia is testing some 122-digit numbers
(2,785-, 2,1614M, 3,555-, 5,575M).

Brent-Montgomery-te Riele numbers. Samuli Larvala is testing them with P-1
and GMP-ECM.

Mersenne numbers. Eric Prestemon found a p21 and a p26 factor of M(1999), 
leaving a c549 cofactor. Our effort with Peter Montgomery to factor M(727),
the smallest Mersenne number with no known prime factor, was unsuccessful.
I did a total of 1567 curves with B1=1e6, 4618 curves with B1=3e6, and Peter
did 140 curves with his special ecmfft code with B1=50e6. This multiplied by
more than two the total ECM effort invested in M(727), which is now 3.5e10
according to Will Edgington table. (The ECM effort is defined as the sum of
the first limit B1 for all curves tried.)

Repunits. Torbjorn Granlund found new factors of 10,259- (p29), 10,265- (p33,
and first known factor apart the trivial one 9), and now also tests numbers 
of the form 10^n+1 (found the first known factor of 10,373+). He has now 
found a total of 147 new factors in 10^n-1 or 10^n+1.

Fibonacci numbers. Ralf Stephan from Germany is testing the 44 remaining
unfactored Fibonacci numbers in range 1400-1500 (125 to 314 digits). He hopes
to have found all factors up to 20 digits, and is now looking for those up to
30 digits.

Cullen and Woodall numbers. Using GMP-ECM, I found a p33 in a 96-digit
cofactor of W(387) [387*2^387-1], that Paul Leyland was about to factor
using his triple-prime variant of the quadratic sieve. Also, Darren Smith
found on March 8 a new Cullen prime, namely C(262419), with 79002 digits,
breaking by a factor of more than 10 the previous record! This is also the
8th largest prime known.

Exotic numbers. On January 30 (but I was informed only after newsletter 1),
Aiichi Yamasaki found a p36 factor of the the numerator from the 215th 
Wolstenholme number 1+1/2+...+1/215. On February 14, he also found a p36
factor of P1*P2*...*P52-P53 = 2*3*...*239-241, where Pn denotes the n-th prime.
Ralf Stephan is factoring Smarandache numbers, of the form 1234567891011121314
or 1413121110987654321 when reversed.

Champions. They were 3 more champions since last newsletter, among which one
was found by GMP-ECM (the p45 from 2,599-, which is the 5th largest factor
ever found by ECM). Richard Brent found on February 22 a p43 of 54,77-, and 
Peter Montgomery on February 13 a p40 in 44,126+. There are now 11 champions
found so far in 1998 (5 by GMP-ECM), for a total of 84 champions.

GMP-ECM top-10 table. The smallest factor found is now a p36 (was a p33 in
February).

Other news. We are finally done with our search for ten consecutive primes
in arithmetic progression. Manfred Toplic stroke again. See [2].

Happy week to all of you,
Paul

[1] The ecmnet home page: http://www.loria.fr/~zimmerma/records/ecmnet.html

[2] http://www.ltkz.demon.co.uk/ar2/10primes.htm