Find all primes p and q such that p divides q2 - 4 and q divides p2 -

1.	Find all primes p and q such that p divides q2 - 4 and q divides p2 - 1.
Answer» Suppose that p ≤ q. Since q divides (p - 1)(p + 1) and q > p - 1 it follows that q divides p + 1 and hence q = p + 1. Therefore p = 2 and q = 3. On the other hand, if p > q then p divides (q - 2)(q + 2) implies that p divides q + 2 or q - 2 = 0. This gives either p = q + 2 or q = 2. In the former case it follows that that q divides (q+2)²-1, so q divides 3. This gives the solutions p > 2; q = 2 and (p; q) = (5; 3).

Find all primes p and q such that p divides q2 - 4 and q divides p2 - 1.

Answer»

Suppose that p ≤ q. Since q divides (p - 1)(p + 1) and q > p - 1 it follows that q divides p + 1 and hence q = p + 1. Therefore p = 2 and q = 3.

On the other hand, if p > q then p divides (q - 2)(q + 2) implies that p divides q + 2 or q - 2 = 0. This gives either p = q + 2 or q = 2. In the former case it follows that that q divides (q+2)²-1, so q divides 3. This gives the solutions p > 2; q = 2 and (p; q) = (5; 3).

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Find all primes p and q such that p divides q2 - 4 and q divides p2 - 1.

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