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Absolutely enormous primes are known these days. It is not possible that all primes are known up to the largest ones one sees mentioned. So how far up is EVERY prime known?
I think they have software that constantly looks for the largest prime. If you run it on your computer and it finds a larger prime, you can get a cash reward.
The primes found by computer search such as GIMPS are almost always "Mersenne primes". These are of the form (2^N) -1, not all N are prime and so computers are used to check which are.
There may also be other primes between Mersenne primes that have been missed.
That's my point. Mersenne primes must be a tiny subset of all primes.
Many cryptographical systems depend on the use of multiples of vast primes. It is not computationally feasible to factorise such numbers, which keeps these systems secure.
Yet if the known vast primes are a minute subset of all primes, and are found by such methods as those used to find Mersenne primes, does this not mean these systems are easier to crack than if all lower primes were known?
It's just as hard to check if any given number is prime wether it is Mersenne or not.
The encryption is based on it being difficult to find which two numbers you used - since the cracker doesn't get to pick probable primes - they have to check all possible factors of the 200 digit multiple.
That's my point. Mersenne primes must be a tiny subset of all primes.
Many cryptographical systems depend on the use of multiples of vast primes. It is not computationally feasible to factorise such numbers, which keeps these systems secure.
Yet if the known vast primes are a minute subset of all primes, and are found by such methods as those used to find Mersenne primes, does this not mean these systems are easier to crack than if all lower primes were known?
What do you mean tiny subset? There are infinitely many Mersenne primes and infinitely many non-Mersenne primes, and both sets are well-ordered.
Yes it's O(1) to check if a number is a mersenne prime - there are only 44 of them!No! It's much easier to prove that a number is a Mersenne prime than to prove the primality of a number of similar size.
Yes it's O(1) to check if a number is a mersenne prime - there are only 44 of them!
I mean't that mersenne primes have no real relevance to ayn fast factoring algorithms that would effect crypto.