DeepSeek R1 evaluations

To find the least odd prime factor of (2019^8 + 1), we start by noting that if a prime ( p ) divides (2019^8 + 1), then (2019^8 \equiv -1 \pmod{p}). This implies (2019^{16} \equiv 1 \pmod{p}), so the order of 2019 modulo ( p ) is 16. By Fermat's Little Theorem, the order must divide ( p - 1 ), hence ( p \equiv 1 \pmod{16} ).

The primes congruent to 1 modulo 16 are 17, 97, 113, 193, etc. We check the smallest primes:

1. Prime 17: Compute (2019 \mod 17):

   • (2019 \equiv 13 \pmod{17})
   • Calculate (13^8 \mod 17):
     • (13^2 \equiv 16 \pmod{17})
     • (13^4 \equiv 16^2 \equiv 1 \pmod{17})
     • (13^8 \equiv 1^2 \equiv 1 \pmod{17})
   • Thus, (2019^8 + 1 \equiv 1 + 1 = 2 \pmod{17}), so 17 is not a factor.

2. Prime 97: Compute (2019 \mod 97):

   • (2019 \equiv 79 \pmod{97})
   • Calculate (79^8 \mod 97):
     • (79^2 \equiv 33 \pmod{97})
     • (79^4 \equiv 33^2 \equiv 22 \pmod{97})
     • (79^8 \equiv 22^2 \equiv 96 \pmod{97})
   • Thus, (2019^8 + 1 \equiv 96 + 1 = 97 \equiv 0 \pmod{97}), so 97 is a factor.

Since 97 is the smallest prime congruent to 1 modulo 16 that divides (2019^8 + 1), the least odd prime factor is (\boxed{97}).