2023 SSMO Team Round Problems/Problem 4
Problem
Find the sum of values for prime 
such that 
Solution
We have that 
, so by Fermat’s Little Theorem,
![\[a^p \equiv a \pmod{p}.\]](http://latex.artofproblemsolving.com/0/e/9/0e927348dc6da2fdd035bfa38f88554c83a98fa8.png)
Thus,
![\[2023^{p^2} + 2^{p^4} \equiv 2023 + 2 \equiv 2024 \pmod{p}.\]](http://latex.artofproblemsolving.com/b/f/5/bf516f24023cebc318dc8a5604c0ab3c81765e2a.png)
In other words, 
. The prime divisors of 2024 are 
, 
, and 
, so the answer is
![\[2 + 11 + 23 = \boxed{36}.\]](http://latex.artofproblemsolving.com/f/2/2/f22209d695635a8c61e5063505e2628d7fb01036.png)
~SMO_Team
