21888242871839275222246405745257275088696311157297823662689037894645226208583 may not necessarily be prime

[jamesray1]

# The prime modulus of the field
field_modulus = 21888242871839275222246405745257275088696311157297823662689037894645226208583
# See, it's prime!
assert pow(2, field_modulus, field_modulus) == 2

https://math.stackexchange.com/questions/2597594/if-2x-bmod-x-2-how-does-that-prove-that-x-a-prime-number-is-prime?noredirect=1#comment5363586_2597594

There is a small probability that it isn't, ~1%.

[mxxun]

Graph of A001567 shows that fraction of Fermat pseudoprimes to base 2 less than n (aka P_2(n)/n) falls with n, with only ~10^5 of those below 10^12. (There are also some proven upper bounds on P_2(n), for example, here). I believe that with field_modulus > 10^76 probability that it is pseudoprime is fairly microscopic.
One can also check that field_modulus is, in fact, prime; this works, or one could implement the underlying algorithms from scratch.
On the other hand, I agree that it is not obvious that pow(2, field_modulus, field_modulus) == 2 is a good test for primality of field_modulus.

[jamesray1]

Well https://www.alpertron.com.ar/ECM.HTM says that 21888242871839275222246405745257275088696311157297823662689037894645226208583 is prime, but we have to assume that it works correctly.