If a finite prime number can be found would a new means of encryption be needed?

Question

Yes I was referring to a solution to the Riemann conjecture if the largest prime could be found and substantiated finite was a poor choice of wording.

Answer 1

All prime numbers are finite. Therefore, no new encryption would be needed if we found another one (which happens roughly once every 5 milliseconds in the routine process of key generation for RSA encryption).

I suspect you meant to ask something else, although I am unable to divine what. Perhaps you should rephrase your question.

Edit: Largest prime? There is no such thing. After all, if there were, that would mean there were only finitely many primes, so we could find a number p which is the product of all the primes. But then, p+1 would not be divisible by any prime numbers, and would thus have to be prime itself -- a contradiction. Therefore there can be no largest prime number.

As for the Riemann hypothesis -- I don't think that it would have any impact on the encryption business. The reason why is that the Riemann hypothesis and its many corollaries have been experimentally confirmed for very large values, so if some encryption code could be made or broken using an algorithm that assumes the Riemann hypothesis, that algorithm could be developed and used even in the absence of a proof. If the Riemann hypothesis is true, the algorithm will still work regardless of whether the hypothesis is proven. The encryption industry have no qualms whatsoever about using unproven mathematical conjectures in their ciphers -- the most famous of these unproven conjectures is that integer factorization is hard. In fact, it might be the case that integer factorization can be done in linear time on a standard computer (although unlikely, since even primality testing is usually O(n³)). This doesn't bother the encryption industry much, because even if it is possible, It's still the case that nobody knows how to do it, and so they can guarantee the safety of your data on that basis.

Answer 2

3 is a finite prime number.

Are you trying to ask something else?

If we find a means to factor numbers to primes in linear time then yes that would impact all modern encryption systems (quantum excluded)