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👉 The longest known mathematical proof is the proof of the infinitude of primes, attributed to Euclid, which has a remarkable length and depth. It begins by assuming there are only finitely many prime numbers, p1, p2,..., pn. Then, Euclid considers the product of all these primes plus one, denoted as N = (p1
p2
...
pn) + 1. This number N is either prime itself or has prime factors not among the original list p1, p2,..., pn. If N is prime, it contradicts the assumption of finitely many primes, as it introduces a new prime not in the list. If N is composite, it must have a prime factor p. However, p cannot be any of the primes p1 through pn, as division by any of these would leave a remainder of 1. Thus, p must be a new prime not in the original list, again contradicting the assumption that the list was complete. This elegant argument demonstrates there are infinitely many primes, showcasing the power and infinite nature of prime numbers in number theory.

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