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Find prime pairs satisfying p squared minus 2q squared equals 1

Check the exceptional even prime first

Two is the only even prime. If an equation changes behavior when one prime is even, test 2 separately, then use modular arithmetic to rule out all odd primes at once.

Problem

Find every ordered pair of prime numbers $(p, q)$ satisfying $p^{2} - 2 q^{2} = 1$ .

Answer

The only pair is

(p, q) = (3, 2)

Substitution gives $9-2(4)=1$.

Step-by-step solution

1. Test the even prime

If $q = 2$ , the equation gives $p^{2} = 9$ , so the prime value is $p = 3$ .

2. Rule out odd primes

Every other prime $q$ is odd, so $q^{2}$ leaves remainder 1 modulo 4. Then $p^{2} = 2 q^{2} + 1$ would leave remainder 3 modulo 4. A square can leave only remainder 0 or 1 modulo 4, so no odd $q$ works.

3. Conclude

The verified pair $(3, 2)$ is therefore unique.

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