Quandaries
and Queries 

HI My name is Antonio . I am a secondary student .I have difficulties in proving the following: The numbers p and 8p^{2} +1 are prime. Prove that the number 8p^{2}+2p+1 is also a prime number. 

Hi Antonio, If p = 3 then 8p^{2} +1 = 73 which is a prime and 8p^{2}+2p+1 = 79 which is also a prime. If p is a prime that is not 3 then division of p by 3 leaves a remainder of either 1 or 2. If the remainder after dividing p by 3 is 1, then the remainder after dividing p^{2} by 3 is also 1. Thus the remainder after dividing 8p^{2} +1 by 3 is 0, and hence 8p^{2} +1 is not a prime. What can you say if the remainder after dividing p by 3 is 2? Claude and Penny 

