Question

Determine all prime numbers a,b,and c for which the expression a^2+b^2+c^ 2-1 is a perfect square.​

Answer 1

Step-by-step explanation:

Modulo 4 all squares are 0 or 1. If all of a,b,c are odd primes (not 2) we have a2+b2+c2−1≡2mod4, which is a contradiction. For similar reasons we cannot have all three of a,b,c being 2, so there must be one or two 2's (which we will assign by default to the earliest variables in alphabetical order). If two, we have 7+c2=d2 with the only solution being c=3 (and d=4).

It remains to check the one-2 case. plop's comment rules out the b,c>3 case, so we are left with a=2, b=3 and the equation 12+c2=d2, whose only solution is c=2 – but this is the same as the previous considered case up to variable reordering.

Hence the only solutions to a2+b2+c2−1=d2 with a,b,c primes are {a,b,c}={2,2,3}.