Factorising Mersenne composites
I have to ask... beyond a simple curiosity, is there any reason behind attempting to factorise a Mersenne number that has already been proven to be composite? I noticed that the user "westicles" successfully found a factor in [URL="http://www.mersenne.org/report_exponent/?exp_lo=973421&full=1"]2[SUP]973,421[/SUP]  1[/URL] today (October 14[SUP]th[/SUP], 2015), even though it's been triplechecked with the LucasLehmer primality test. Of course, the factor found in this case is 88 bits long, so is definitely an impressive find, but is there a good reason for looking for the factors in such composites?

[QUOTE=CuriousKit;412716]I have to ask... beyond a simple curiosity, is there any reason behind attempting to factorise a Mersenne number that has already been proven to be composite? I noticed that the user "westicles" successfully found a factor in [URL="http://www.mersenne.org/report_exponent/?exp_lo=973421&full=1"]2[SUP]973,421[/SUP]  1[/URL] today (October 14[SUP]th[/SUP], 2015), even though it's been triplechecked with the LucasLehmer primality test. Of course, the factor found in this case is 88 bits long, so is definitely an impressive find, but is there a good reason for looking for the factors in such composites?[/QUOTE]
For the fun of it. Some people like to move Mersenne numbers from the doublechecked to hasaknownfactor state. Others like the even tougher challenge of moving Mersenne numbers from the hasaknownfactor state to allfactorsknown state. 
To add to Prime95's post. The entire project is also for the fun of it. So it doesn't matter if someone wants to factor, or run an LL, or neither; the world will still turn as usual and wars will still be fought as usual.

For fun... heh, I can buy that! Although I'd hope we can stop the wars at least.

the main use I could find for it is in trying to eliminate factors for later exponents as they can't share factors unless the exponent itself is composite). Of course this probably would never work.

Well, in July I found the prime factor 782521855299947974696932851410613860657 of [URL="http://www.mersenne.org/report_exponent/?exp_lo=1864739&full=1"]M1864739[/URL]. I've been running the P1 algorithm on Mersenne numbers with [b]known[/b] factors for numbers less than M2000000. The idea is to try to find a "probably completely factored" Mersenne number, where that term means a product of primes and a probable prime.
I had several successes so far: [CODE] (2^17907431)/(146840927*158358984977*3835546416767873*20752172271489035681) = PRP539014 (2^7501511)/(429934042631*7590093831289*397764574647511*8361437834787151*17383638888678527263) = PRP225744 (2^6759771)/(1686378749257*7171117283326998925471) = PRP203456 (2^5765511)/4612409/64758208321/242584327930759 = PRP173528 (2^4884411)/(61543567*30051203516986199) = PRP147012 (2^4403991)/(16210820281161978209*31518475633*880799) = PRP132538 (2^2700591)/540119/6481417/7124976157756725967 = PRP81265 [/CODE] The number at the right of "PRP" is the number of digits of that probable prime. 
[QUOTE=CuriousKit;412716]I have to ask... beyond a simple curiosity, is there any reason behind attempting to factorise a Mersenne number that has already been proven to be composite? I noticed that the user "westicles" successfully found a factor in [URL="http://www.mersenne.org/report_exponent/?exp_lo=973421&full=1"]2[SUP]973,421[/SUP]  1[/URL] today (October 14[SUP]th[/SUP], 2015), even though it's been triplechecked with the LucasLehmer primality test. Of course, the factor found in this case is 88 bits long, so is definitely an impressive find, but is there a good reason for looking for the factors in such composites?[/QUOTE]
Welcome to "Wagstaff's stamp collection" 
SSW isn't the only person with a stamp collection. I have one and so do you though, to be fair, I'm the one looking after your stamp albums for the moment.
Nothing wrong with stamp collection as long as it's clear at least to the collector that that is the activity being undertaken. For the avoidance of doubt, I'm pretty sure that you agree. 
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