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 Peter Hackman 2008-08-14 07:02

lots of large primes

I going over my QS hobby programs I've been running two versions, one with multipliers, one without.

Typically my runs have been on 50-digit numbers (as I've indicated elsewhere my Python programs are very slow), sieving interval [-10**5, 10**5], factorbase 3000, lowest prime sieved on 37, large prime bound 200*pmax. I've sieved until smooths+0.07*partials exceeds 0.96*length of factor base. This is safe, not to say pessimistic, as I usually get
something like 100-200 relations.

However, in some cases, especially when the multiplier is small, I get an enormous amount of redundance, slowing things down considerably. In other words, in these cases I get many more matches
than expected.
I suppose I could modify my program
to deal with this phenomenon. Just curious whether there is some simple explanation for it.

 jasonp 2008-08-14 12:34

Hard to say without knowing more about your implementation, but even with SIQS the number of duplicates is not expected to be large. I would suspect a bug, or at least inadvertent selection of the same polynomials multiple times. The other possibility is that your factor base is too small, so that if you only allow one large prime above the factor base bound then that large prime occurs often, in many relations.

 Peter Hackman 2008-08-15 14:26

Perhaps the example below helps clarify my observations/questions.
I've run it with varying small primes bounds (but always the same tolerance term, 2*int(log(pmax))
and factor base sizes. In my program the ideal seems to be a little above 2000, giving a number of factorizations very close to, sometimes slightly smaller than, the size of the factor base.
Each polynomial gives rise to 10 plus or minus 3 trials and I believe the success rate is something like 85-90% (I've no idea how many candidates I'm missing!). What strikes me is the much larger percentage of matches in the first case, and this is a recurring observation.

N=27656492312230575123821463511979706363619541782727
smoothness bound 45000,
smalles prime sieved on: 37.

multiplier 3:

1793 polynomials
1311 smooths
large prime factorizations 13132
matches 1774
number of relations: 785

multiplier 1:

1575 polynomials
1216 smooths
large prime factorizations: 14943
matches: 1277
number of relations: 170

This is really a different topic, but I don't get duplicate polynomials. I generate the a=q*q coefficients by picking a number at random in a small neighborhood of the ideal size and look for the next larger prime number q
with q&3=3.
I keep a list of the q's and check for duplicates (I've tried hashing, too). For some reason this approach seems to be faster than stepping outwards from the ideal! I'm sure there are better ways, but I haven't seen any discussion of this in the literature.

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