Problem:
Which of the following expressions is never a prime number when p is a prime number?
Answer Choices:
A. p2+16
B. p2+24
C. p2+26
D. p2+46
E. p2+96
Solution:
If p=3, then p2+26=35=5⋅7. If p is a prime number other than 3, then p=3k±1 for some positive integer k. In that case
(C)p2+26​=(3k±1)2+26=9k2±6k+27=3(3k2±2k+9)
is a multiple of 3 and is not prime. The smallest counterexamples for the other choices are 52+16=41,72+24=73,52+46=71, and 192+96=457.
The problems on this page are the property of the MAA's American Mathematics Competitions