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2006-03-23, 01:18   #3
R.D. Silverman

Nov 2003

26·113 Posts

Quote:
 Originally Posted by ET_ Playing with my applet, i found out this beautiful result: M1000000000000000000000000000000000000000000000000000000001059 has a factor: 40000000000000000000000000000000000000000000000000000000042361 The factor being a 62-digit prime (about 204.62 bit) obtained with trial-factoring software. Geee, it's fun! Luigi
If people would only learn a little mathematics, this kind of silliness would
stop. I keep telling people: do a little math BEFORE computing. But noone
seems to listen. I speculate that this is because math is hard, computing is easy, and participants herein can't be bothered doing anything that is *hard*.

The reward that comes from doing something HARD is a lot greater than
doing something EASY.

It is a *TOTALLY trivial* matter to find very large factors of very very large
Mersenne numbers. I will give a hint: Let p be a prime that is 3
mod 4 such that 2p+1 is also prime. Now consider the Mersenne
number M_p. Think 'quadratic reciprocity' and 'Euler's Theorem'.

Note that this 'factor' is discovered without any "trial division" at all.

What *would* be impressive would be finding a 62 digit factor of a
relatively small Mersenne number. (say) p < 2000.

And factors larger than 62 digits of Mersenne numbers have been found.
Quite a few. Look at 2^683-1, 2^727-1, and 2^811-1, for example.