2022/035) Find the smallest prime factor of $2019^8 + 1$

let the prime be p

we have $2019^8 \equiv -1 \pmod p$ 

or $2019^{16} \equiv 1 \pmod p$

so $2019^{16k} \equiv 1 \pmod p$

as per Formats Little theorem p shall be of the form 16k + 1.

any number less than 16 is not possible because  $2019^8 \equiv -1 \pmod p$ and $16=2^4$ and $8=2^3$ so any factor of 16 shall not give a power =1.

so p = 16 *2 + 1 = 33 (not a prime) or 16 * 3 + 1= 49( not a prime) or 16 * 4 + 1 = 65( not a prime) or 16 * 5 + 1 = 81 (not a prime) or 16 *6  + 1 = 97.

so we check for 97 

now $2019 \equiv - 18 \pmod {97}$ 

so $2019^2 \equiv 324 \pmod {97}$

or   $2019^2 \equiv 33  \pmod {97}$

so $2019^4 \equiv 33^2  \pmod {97}$

or $2019^4 \equiv 22   \pmod {97}$

or $2019^8 \equiv 484  \pmod {97}$

or $2019^8 \equiv -1  \pmod {97}$

or or $2019^8 + 1\equiv 0   \pmod {97}$

so smallest prime factor = 97