I do not know what your are asking but there is an elegant indirect proof of infinitude of primes that I know; it comes from Euclid. Assume, that there are finitely many primes, . Now form a number,Originally Posted byLazarath

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Buy the fundamental theorem of arithmetic this number factors into primes number and thus is divisible by a prime number. But non of these prime numbers from divide . Because if then but since then

which is not possible. Thus, there cannot be finitely many prime numbers.