The number of primes is infinite

Reference: Proofs from THE BOOK (Third Edition) (Martin Aigner and Gunter M. Ziegler)

Suppose the set of primes, \mathcal{P} is finite and p is the largest prime. We consider the so-called Mersenne number 2^{p}-1, and show that any prime factor q of 2^{p}-1 is bigger than p, which will yield the desired conclusion. Let q be a prime dividing 2^{p}-1, so we have 2^{p} \equiv 1 \pmod {q}.

Since p is prime, this means that the element 2 has order p in the multiplicative group Z_{q} { \\ \{0\}} of the field Z_{q}. This group has q-1 elements. By Lagrange’s theorem, we know that the order of every element divides the size of the group, that is, we have p|q-1, and hence, p<q.

More later,

Nalin Pithwa

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