The ’standard’ proof of the infinitude of primes implies that: Given primes p1, p2, …, pn, multiply them, add 1, and factor the result. All its prime factors a

The ’standard’ proof of the infinitude of primes implies that: Given primes p1, p2, …, pn, multiply them, add 1, and factor the result. All its prime factors are new primes. For example, given {5,7,11}, the product is 385. Adding one yields 386. Factoring that learns you that 386 = 2 × 193. Both are primes not in the original list. If you start with {}, the series you get is {} ⇒ 2 = 2 {2} ⇒ 3 = 3 {2,3} ⇒ 7 = 7 {2,3,7} ⇒ 43 = 43 {2,3,7,43} ⇒ 1807 = 13 × 139 {2,3,7,43,13,139} ⇒ 3263443 = 3263443 (https://oeis.org/A126263 https://oeis.org/A126263; i think it doesn’t generate all primes)