Euclid's theorem
Proof by construction

Filip Saidak gave the following proof by construction, which does not use reductio ad absurdum or Euclid's lemma (that if a prime p divides ab then it must divide a or b).

Since each natural number greater than 1 has at least one prime factor, and two successive numbers n and (n + 1) have no factor in common, the product n (n + 1) has more different prime factors than the number n itself. So the chain of pronic numbers: 1×2 = 2 {2}, 2×3 = 6 {2, 3}, 6×7 = 42 {2, 3, 7}, 42×43 = 1806 {2, 3, 7, 43}, 1806×1807 = 3263442 {2, 3, 7, 43, 13, 139}, · · · provides a sequence of unlimited growing sets of primes.
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