Primality Testing and Factoring Algorithms

[raman22feb1988]

3^607-1 only that trivial factor of 2 had been known before.
The 290 digit composite number has been factored completely
by that NFS@Home team (under BOINC resources - Berkeley Open
Infrastructure for Network Computing), into 3 prime factors as

prp85 factor: 2086414114912368934845698889752452759455632789377648284275990116236503745140234728069

prp96 factor: 120463408255967053578135908158245347326036064518355187873832329408825869652943374488400197042063
prp110 factor: 81529721835493524725988100443970680738653595633187211135428836315200448881013673156176891838179958882238447519


by using that snfs algorithm. This number was one of that odd perfect number roadblocks, at 578 digits
This is second largest ever factorization by using that Special Number Field Sieve,
after that 2^1039-1, 2^1127-1 (2^1183-1 had been in due, under progress by using that SNFS, but is now completed up).

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2^1237-1 had been the Mersenne number with no known factors at all, after that 2^1061-1
Upon November 2nd 2010, that Arjen Lenstra, Peter Montgomery, Joppe Bos, Thorsten Kleinjung found out a 70 digit prime
factor from that number by using that Elliptic Curve Method.
p70 = 2538207129840687799335203259492870476186248896616401346500027311795983
That 303 digit cofactor is still composite, yet as of now.

Now after that 2^1277-1 is of such the case with no known prime factors for that number, followed by
the next one that is only at 2^1619-1. (2^1279-1) is prime.

Recently found out first factors for these numbers, at that way, rather within that case

p55 factor = 5198395892876421104415109549087087419559080537214372111 from

M2269
p56 factor = 10788426156438350117334292343137689257142387557947087583 from
M1657
p57 factor = 112493750443412941745410571996247741731544451845539488817 from
M1669

This is third largest factor that has been found out by using that ECM algorithm, ever,
after that two 73 digit prime factors from that two Mersenne numbers 2^1181-1, 2^1163-1,
by this same team, by using that playstation console cluster, earlier this year.
only actually
ny nuch
stztz tztz stz tz

[DOSGuy]

I now have 3 primes in the Top 5000 Largest Known Primes at the same time!

3243 * 2^666405 + 1 (200 612 digits) is 4747th (entered at 4744th)
8397 * 2^665810 + 1 (200 433 digits) is 4768th (entered at 4756th)
4701 * 2^663044 + 1 (199 600 digits) is 4893rd (entered at 4765th)

These are actually my 5th, 6th, and 7th prime numbers. After I discovered my second prime in January 2010, I spent the rest of the year focusing on sieving, but I decided to get back on the Top 5000 when my second prime fell off the list. I discovered a prime in December 2010 (ranked 4947th at the time) that quickly fell of the list, so I didn't bother mentioning it. I found a much larger one in February 2011 (ranked 4597th at the time, down to 5782nd now). Then, I discovered three primes in the same week in March, which have finally been verified today.