Fun with factorization

I'm pleased to report that 253262965868307257488778209946038131361 is a factor to P134-1, where P134 is a prime factor of M757=2^757-1 from NFSNET and is also given below. The desire to factor "prime minus 1" stems from proving primality as well as interesting cycles within the Galois field generated by the prime. The computation took 3.27 days using the MPQS routine in PARI/GP. The final linear algebra step reported by GP is: MPQS: starting Gauss over F_2 on 40260 distinct relations MPQS: Gauss done: kernel has rank 1263, taking gcds... MPQS: splitting N after 70 kernel vectors MPQS: time in Gauss and gcds = 1779770 ms Thus, the complete factorization of P134-1 is 2 * 757 * 31771 * 225975587 * 426665371159 * 2901513458813 * 253262965868307257488778209946038131361 * 7052139211996899091204255207082765933446054813418773253. The product of the last two factors, the 94-digit number 1786045692546521893242296017081938219863268932108739868225086786180708765562719633225087287333, was the hard thing to factor. Note that 757 occurs as a factor, which is not a coincidence. The penultimate factor of M757 is P79, and P79-1 factors without too much trouble: P79-1 = 2 * 2 * 2 * 2 * 13 * 757 * 1229897 * 54286614780461303 * 1380765008073205793 * 394201037959710813849206239443089. Again 757 occurs as a factor. Just to repeat the NFSNET result: 2^757-1 = 9815263 * 561595591 * P79 * P134, where P79 = 5722137022002067824248227975095857749151312827809388406962346253182128916964593 and P134 = 24033821640983508088736273403005965446689002356344332130565066643193813901119771090424269412054543072714914742665677774247325292327559