Proofs of Existence and Optimality · Constructive Proofs of Existence
Lesson 5
It is not difficult to use the code provided above to find the first \(n\) for which \(p(n)\) is not prime.
from itertools import count
from sympy import isprime
def p(n):
return n ** 2 - n + 41
print(next(n for n in count() if not isprime(p(n))))41
If you look closely, it is not so surprising that \(p(41)\) is composite: after all, \(p(41)>41\) and \(p(41)\) is divisible by \(41\). What is surprising is that for all integers \(0 \le n < 41\), the number \(p(n)\) is prime.
For the curious 🤓
It is known that the number \(41\) is the largest Euler's lucky number. These are positive integers \(k\) such that the value of the polynomial \(n^{2}-n+k\) is prime for all integers \(0 \le n < k\).