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I'm sorry to say that your code is likely to have essentially zero use. Primes used for cryptography (e.g., RSA), are on the order of 2,048 and 4,096 bits of length, or respectively roughly 616 and 1,233 digits long. Algorithms already exist to rapidly find (random) primes of this size, and unless you've broken new ground in number theory, your approach is ...


0

To answer your question about the size of primes: The current recommendation for RSA keys recommends using two 1024-bit keys. That means they're somewhat larger than $10^{300}$; Diffie-Hellman on the integers modulo $p$ is normally used with a single prime of at least 2048 bits (so, larger than $10^{600}$!) Even worse, these numbers tend to grow over time; ...


1

This question is really broad. I'll try to answer in a few sentences. Of course, $\mathbb Z$ in its widely accepted definition has infinitely many primes. This means: the properties people usually expect from something we may rightly call "the integers" already imply that this thing contains infinitely many primes. Hence it is impossible to keep everything ...



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