Euler’s prime-producing polynomial revisited

Euler’s polynomial f ( n ) = n 2 + n + 41 is famous for producing 40 different prime numbers when the consecutive values 0, 1, …, 39 are substituted: see Table 1. Some authors, including Euler, prefer the polynomial f ( n − 1) = n 2 − n + 41 with prime values for n = 1, …, 40. Since f (− n ) = f ( n − 1), f ( n ) actually takes prime values (with each value repeated once) for n = −40, −39, …, 39; equivalently the polynomial f ( n − 40) = n 2 − 79 n + 1601 takes (repeated) prime values for n = 0, 1, …, 79.