Best Work for Finding Primes
Hello,
Right now I have my computers "trial factoring to low limits" by using the "LucasLehmer test." I don't know what any of that means to be quite honest but I just wanted to make sure I'm doing the right thing so I can help test a new prime! Sorry to be a bother but just making sure. Is there anything else I should be doing? Thanks! 
Welcome
Trial factoring (TF) is a preliminary step that eliminates possible candidates by finding small factors. Factor found = not prime.
For proving a prime you must run "first time Lucas Lehmer tests" (LL) which you seem to have demanded from Prime95. But if your PC is relatively slow PrimeNet will assign less demanding work (for example TF). Could you add your PC's hardware? Last but not least: Welcome to the fun world of finding primes and factors. 
Just to add to what Brain writes:
Trial factoring, and LucasLehmer testing, are two necessary types of work. They are not the same, and there are other work types as well. You can set a work type preference if you wish on your PrimeNet account to choose what you want to do, or you can simply select the "what makes sense" option which will make PrimeNet decide what work to give you on the basis of your hardware's particular strengths plus what the project most needs at the moment. If you want to choose your own work type, and if you want a chance of discovering a mersenne prime yourself then, as Brain says, first time LL testing is the way to go. (DC, which is doing an LL test that someone else has already done as a double check, provides a smaller chance of discovering a prime too.) An LL test will take weeks or months, though. Factoring work is faster, just as important to the project, and much more likely to give a positive result (factor found); but it will not lead to you being the discoverer of a new prime. 
[QUOTE=BrianE;303077]Just to add to what Brain writes:
Trial factoring, and LucasLehmer testing, are two necessary types of work. They are not the same, and there are other work types as well. You can set a work type preference if you wish on your PrimeNet account to choose what you want to do, or you can simply select the "what makes sense" option which will make PrimeNet decide what work to give you on the basis of your hardware's particular strengths plus what the project most needs at the moment. If you want to choose your own work type, and if you want a chance of discovering a mersenne prime yourself then, as Brain says, first time LL testing is the way to go. (DC, which is doing an LL test that someone else has already done as a double check, provides a smaller chance of discovering a prime too.) An LL test will take weeks or months, though. Factoring work is faster, just as important to the project, and much more likely to give a positive result (factor found); but it will not lead to you being the discoverer of a new prime.[/QUOTE] basically what is being pointed out here is that gimps does not do TF by LL. 
[QUOTE=science_man_88;303083]basically what is being pointed out here is that gimps does not do TF by LL.[/QUOTE]
Okay thank you. When I go to test > status it says that both workers are doing LucasLehmer test so from what I now understand, that's good! I have a 2.4 Ghz Intel Core 2 Duo. I think I'm good then if I'm not mistaken. 
[QUOTE=science_man_88;303083]basically what is being pointed out here is that gimps does not do TF by LL.[/QUOTE]
As well I selected the choose the work for me option since I thought it would be best. 
[QUOTE=science_man_88;303083]basically what is being pointed out here is that gimps does not do TF by LL.[/QUOTE]
It's not GIMPS, it is what an LL test tells you: Either the number is prime or it's composite, but if composite gives no indication as to what the factors are. (Although one might conjecture that the actual nonzero residue could potentially help in finding the factors). It is worth checking for factors below 2^73. GIMPS typically searches up to 2^71, because of an administrative cockup. 
[QUOTE=davieddy;303130](Although one might conjecture that the actual nonzero residue
could potentially help in finding the factors).[/QUOTE] funny you say that because I've brought a way up before, I believe. it's just I'm almost positive it's more expensive than TF. PM me if you want details. 
EWMayer has explicitly said that the (full) residue can be used to effect a primality test on the cofactor after finding a first factor.

[QUOTE=Dubslow;303155]EWMayer has explicitly said that the (full) residue can be used to effect a primality test on the cofactor after finding a first factor.[/QUOTE]
the first m*y+z y>0 can be used so for 2^111 that means (194^22)%(2^111) = 788 can be used. 788%23 = 6 =((194%23)^22)%23 since these equal 23 is a factor of m*y at least and in this case a factor of m. 
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