{{{id=1|
def pollard_pm1(N,B=0):
    if not B: B=ceil(sqrt(N))
    a = Integers(N).random_element()
    b = a
    for ell in primes(B):
        q = 1
        while q < N:
            q *= ell
        b = b^q
        d = gcd(b.lift()-1,N)
        if d == N: return 0
        if d > 1: return d
    return 0
    
def random_unsafe_prime(bits):
    while true:
        a=randint(0,bits)
        b=randint(0,floor((bits-a)/log(3,2)))
        c=randint(0,floor((bits-a-floor(b*log(3,2)))/log(5,2)))
        d=floor((bits-a-floor(b*log(3,2))-floor(c*log(5,2)))/log(7,2))
        p = 2^a*3^b*5^c*7^d+1
        if is_prime(p): return p
///
}}}

{{{id=5|
factor(random_unsafe_prime(500)-1)
///
2^166 * 3^152 * 5^37 * 7^3
}}}

{{{id=2|
p1=random_unsafe_prime(500)
p2=random_prime(2^500,2^499)
print p1,p2
time print pollard_pm1(p1*p2)
///
4914170138219337312763129025659942324571395707443434262754113983316284499573057205945536803074309022946558860993484794485843208389626167296000000000001 1502372478319849667842662388844675307190055917420021641356062402891154329644640734465578565732879594697319287113459020853626896806657678212714228751677
4914170138219337312763129025659942324571395707443434262754113983316284499573057205945536803074309022946558860993484794485843208389626167296000000000001
Time: CPU 0.02 s, Wall: 0.02 s
}}}

{{{id=6|
time factor(p1*p2, verbose=4)        # this could take a *very* long time
///
WARNING: Output truncated!  


IFAC: cracking composite
	7382913969441983992525588541623489823073874539750544223577885638455251673228165565592372095658051401533859524397866454580732448292371318770760453574996270016552870651818862907403626660509045814945630877973187658446614445792587816461463088974342122119093423988734940429371052553322700233604714228751677
IFAC: checking for pure square
IFAC: checking for odd power
IFAC: trying Pollard-Brent rho method
Rho: searching small factor of 1000-bit integer
Rho: using X^2+1 for up to 49152 rounds of 32 iterations
Rho: time =   2198 ms,	24576 rounds
Rho: fast forward phase (8192 rounds of 64)...
Rho: time =   1011 ms,	32772 rounds, back to normal mode
Rho: time =    596 ms,	40960 rounds
Rho: time =    597 ms,	Pollard-Brent giving up.
IFAC: trying Shanks' SQUFOF, will fail silently if input
      is too large for it.
IFAC: trying Lenstra-Montgomery ECM
ECM: working on 64 curves at a time; initializing for up to 388 rounds...
ECM: time =      0 ms
ECM: dsn = 12,	B1 = 1800,	B2 = 198000,	gss =  128*420
ECM: time =   2009 ms, B1 phase done, p = 1801, setting up for B2
ECM: time =     34 ms, entering B2 phase, p = 2017
ECM: time =   1236 ms
ECM: dsn = 14,	B1 = 2200,	B2 = 242000,	gss =  128*420
ECM: time =   2391 ms, B1 phase done, p = 2203, setting up for B2
ECM: time =     34 ms, entering B2 phase, p = 2413
ECM: time =   1457 ms
ECM: dsn = 16,	B1 = 2700,	B2 = 297000,	gss =  256*420
ECM: time =   2893 ms, B1 phase done, p = 2707, setting up for B2
ECM: time =     35 ms, entering B2 phase, p = 2917
ECM: time =   1767 ms
ECM: dsn = 18,	B1 = 3250,	B2 = 357500,	gss =  256*420
ECM: time =   3398 ms, B1 phase done, p = 3251, setting up for B2
ECM: time =     35 ms, entering B2 phase, p = 3461
ECM: time =   2064 ms
ECM: dsn = 20,	B1 = 4000,	B2 = 440000,	gss =  256*420
ECM: time =   4171 ms, B1 phase done, p = 4001, setting up for B2
ECM: time =     35 ms, entering B2 phase, p = 4211
ECM: time =   2485 ms
ECM: dsn = 22,	B1 = 4850,	B2 = 533500,	gss =  256*420
ECM: time =   4981 ms, B1 phase done, p = 4861, setting up for B2
ECM: time =     36 ms, entering B2 phase, p = 5077
ECM: time =   2989 ms
ECM: dsn = 24,	B1 = 5800,	B2 = 638000,	gss =  256*420
ECM: time =   5890 ms, B1 phase done, p = 5801, setting up for B2
ECM: time =     35 ms, entering B2 phase, p = 6011
ECM: time =   3578 ms
ECM: dsn = 26,	B1 = 7100,	B2 = 781000,	gss =  256*420
ECM: time =   7152 ms, B1 phase done, p = 7103, setting up for B2
ECM: time =     37 ms, entering B2 phase, p = 7313
ECM: time =   4400 ms
ECM: dsn = 28,	B1 = 8700,	B2 = 957000,	gss =  256*420
ECM: time =   8643 ms, B1 phase done, p = 8707, setting up for B2
ECM: time =     36 ms, entering B2 phase, p = 8921
ECM: time =   5399 ms
ECM: dsn = 30,	B1 = 10600,	B2 = 1166000,	gss =  512*420
ECM: time =  10456 ms, B1 phase done, p = 10601, setting up for B2
ECM: time =     37 ms, entering B2 phase, p = 10813
ECM: time =   6593 ms
ECM: dsn = 32,	B1 = 12900,	B2 = 1419000,	gss =  512*420
ECM: time =  12605 ms, B1 phase done, p = 12907, setting up for B2

...

ECM: time =   7992 ms
ECM: dsn = 34,	B1 = 15700,	B2 = 1727000,	gss =  512*420
ECM: time =  15279 ms, B1 phase done, p = 15727, setting up for B2
ECM: time =     37 ms, entering B2 phase, p = 15937
ECM: time =   9680 ms
ECM: dsn = 36,	B1 = 19000,	B2 = 2090000,	gss =  512*420
ECM: time =  18231 ms, B1 phase done, p = 19001, setting up for B2
ECM: time =     40 ms, entering B2 phase, p = 19217
ECM: time =  11657 ms
ECM: dsn = 38,	B1 = 23200,	B2 = 2552000,	gss =  512*420
ECM: time =  22242 ms, B1 phase done, p = 23201, setting up for B2
ECM: time =     38 ms, entering B2 phase, p = 23411
ECM: time =  14157 ms
ECM: dsn = 40,	B1 = 28000,	B2 = 3080000,	gss =  512*420
ECM: time =  26669 ms, B1 phase done, p = 28001, setting up for B2
ECM: time =     38 ms, entering B2 phase, p = 28223
ECM: time =  16992 ms
ECM: dsn = 42,	B1 = 34500,	B2 = 3795000,	gss =  512*420
ECM: time =  32707 ms, B1 phase done, p = 34501, setting up for B2
ECM: time =     38 ms, entering B2 phase, p = 34717
ECM: time =  20772 ms
ECM: dsn = 44,	B1 = 43000,	B2 = 4730000,	gss = 1024*420
ECM: time =  40660 ms, B1 phase done, p = 43003, setting up for B2
ECM: time =     38 ms, entering B2 phase, p = 43219
ECM: time =  25744 ms
ECM: dsn = 46,	B1 = 53800,	B2 = 5918000,	gss = 1024*420
ECM: time =  50559 ms, B1 phase done, p = 53813, setting up for B2
ECM: time =     39 ms, entering B2 phase, p = 54023
ECM: time =  31914 ms
ECM: dsn = 48,	B1 = 67750,	B2 = 7452500,	gss = 1024*420
ECM: time =  63462 ms, B1 phase done, p = 67751, setting up for B2
ECM: time =     39 ms, entering B2 phase, p = 67961
ECM: time =  39813 ms
ECM: dsn = 50,	B1 = 85300,	B2 = 9383000,	gss = 1024*420
ECM: time =  79610 ms, B1 phase done, p = 85303, setting up for B2
ECM: time =     39 ms, entering B2 phase, p = 85517
ECM: time =  49695 ms
ECM: dsn = 52,	B1 = 107400,	B2 = 11814000,	gss = 1024*420
ECM: time =  99948 ms, B1 phase done, p = 107441, setting up for B2
ECM: time =     40 ms, entering B2 phase, p = 107657
ECM: time =  62026 ms
ECM: dsn = 54,	B1 = 135400,	B2 = 14894000,	gss = 1024*420
ECM: time = 125553 ms, B1 phase done, p = 135403, setting up for B2
ECM: time =     39 ms, entering B2 phase, p = 135617
ECM: time =  77534 ms
ECM: dsn = 56,	B1 = 170800,	B2 = 18788000,	gss = 1024*420
ECM: time = 157814 ms, B1 phase done, p = 170801, setting up for B2
ECM: time =     39 ms, entering B2 phase, p = 171013
ECM: time =  97052 ms
ECM: dsn = 58,	B1 = 215400,	B2 = 23694000,	gss = 1024*420
ECM: time = 198626 ms, B1 phase done, p = 215417, setting up for B2
ECM: time =     39 ms, entering B2 phase, p = 215651
ECM: time = 121396 ms
ECM: dsn = 60,	B1 = 271400,	B2 = 29854000,	gss = 1024*420
ECM: time = 249676 ms, B1 phase done, p = 271409, setting up for B2
ECM: time =     39 ms, entering B2 phase, p = 271637
ECM: time = 151808 ms
ECM: dsn = 62,	B1 = 341500,	B2 = 37565000,	gss = 1024*420
ECM: time = 313242 ms, B1 phase done, p = 341501, setting up for B2
ECM: time =     41 ms, entering B2 phase, p = 341713
}}}

{{{id=7|

///
}}}