Checking primability

[drmurat]

I want to check giant numbers as prime or as twin prime. Maybe a sample can be 10 ^ 300.000.000 + N. Which tools I Must Use. What must I do

[paulunderwood]

Originally Posted by drmurat

I want to check giant numbers as prime or as twin prime. Maybe a sample can be 10 ^ 300.000.000 + N. Which tools I Must Use. What must I do

• Get one of these top500 computers
  
• Upgrade you household electricity system to handle up to 30 megawatts
  
• Build a power station nearby
  
• Use PFGW?

[drmurat]

Originally Posted by paulunderwood

• Get one of these top500 computers
  
• Upgrade you household electricity system to handle up to 30 megawatts
  
• Build a power station nearby
  
• Use PFGW?

I Hope it is done by computer program or application

[paulunderwood]

Originally Posted by drmurat

10 ^ 300.000.000 + N

Even after you have done what I said to find a PRP or PRP twin, you would need to live for millions of years and utilise the power of a star in order to prove them prime.

[drmurat]

Originally Posted by paulunderwood

Even after you have done what I said to find a PRP or PRP twin, you would need to live for millions of years and utilise the power of a star in order to prove them prime.

any easy way?

[paulunderwood]

Originally Posted by drmurat

any easy way?

Yes! join GIMPS and look for 100 million digit Mersenne primes, although this a daunting task at the moment. To test a single Mersenne at this size takes a few weeks on a Radeon VII GPU.

[drmurat]

Originally Posted by paulunderwood

Yes! join GIMPS and look for 100 million digit Mersenne primes, although this a daunting task at the moment. To test a single Mersenne at this size takes a few weeks on a Radeon VII GPU.

Any factorization site? İt can be helpful

[paulunderwood]

Originally Posted by drmurat

Any factorization site? İt can be helpful

The GIMPS softwares prime95/mprime and gpuOwl do P-1. There are some other softwares specifically for division tests. This site has some details: mersenne.ca

[VBCurtis]

Originally Posted by drmurat

any easy way?

What is easy and what is hard depends on the form of the number you wish to test. Numbers of no special form are hopeless to prove prime larger than about 100,000 decimal digits, while numbers with special forms such as Mersennes can reasonably be proven up to about 300,000,000 digits (and this bound grows routinely as software is developed).

Projects like mersenne.org exist to take advantage of the special form to look for record-largest-known primes; there are other forms nearly as quick to check, such as k * 2^n -1.

The problem with your original question is the "+N". Arbitrary numbers added to a large power are not a special form. If you stick to +1 or -1, you can find a quite-large prime; though 300M decimal digits such as you suggest would be quite a lengthy search for an individual. Each check of an individual number would be not that hard- but each check has an astronomically low chance to find a prime. How many times will you play the lottery?

[Dr Sardonicus]

Just for practice, I used Pari-GP to find n such that 10^300000000 + n has no microscopically tiny prime factors p <= primelimit, which at the time happened to be a bit over 60000000. I let it run until it gave two sets of consecutive odd numbers.

(The possible factors here are so small, I took the remainder e = 300000000%(p-1) and looked at Mod(10,p)^e + n instead of trying to calculate Mod(10,p)^300000000.)

I'm sure that few, if any, of the values 10^300000000 + n are prime for these 41 n's. I'm not sure how many could be knocked out by a reasonable amount of further simpleminded checking for small factors.

[3,63,133,141,159,171,187,189,289,313,319,347,427,453,469,507,589,613,647,687,691,697,711,717,751,817,827,837,841,913,947,973,999,1029,1041,1047,1051,1071,1303,1307,1309]

[kriesel]

Originally Posted by drmurat

I want to check giant numbers as prime or as twin prime. Maybe a sample can be 10 ^ 300.000.000 + N. Which tools I Must Use. What must I do

First, study the subject. Learn the applicable number theory, and the relevant software offerings and their capabilities and limitations.

~10^300000000+ small N ~2^996,578,429-c. If c= 1, it's a Mersenne number. Such large Mersenne numbers can be done more quickly than numbers of more general form, because of the particular special form, but scaling even with fastest available implementations of fastest known appropriate algorithms on fastest consumer-price hardware takes months for primality testing one candidate.
For example, after a week or more of trial factoring and P-1 factoring, primality testing M993112609 using a recent fairly speed-optimized version of gpuowl on a Radeon VII with 111% overclocked gpu ram on Windows takes ~139 days with electrical power limit reduced for economic reasons, so may on Linux and 120% gpu ram overclock take ~3.5 to 4 months. Scaling up run time from there to M996578447 which is the next prime exponent larger than 996578429 would be ~2.1 power of the exponent ratio, or ~1.00734 times as long, adding about another day. But M996578447 happens to have already been factored.

https://www.mersenneforum.org/showthread.php?t=24607 is almost entirely specific to Mersenne prime hunting. If you are determined to pursue numbers very close to integer powers of ten, you'll need different software, and more hardware, power, and patience.