Prime!
Number Theory
Level
3
P and Q are prime numbers such that the numbers (P+Q) and (P+7Q) are both perfect squares Find the value of P.
The answer is 2.
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Let P + Q = k^2 and P + 7Q = m^2. Subtracting, 6Q = (m - k)(m + k). Since both factors must have the same parity (since their sum is even), and their product is even, each factor is even, which means that 6Q is divisible by 4. So Q must be even, and being prime, Q = 2. Then P = k^2 - 2 and P = m^2 - 14.Subtracting, m^2 - k^2 = 12. So m = 4, and k = 2. Either way, P = 2. Ed Gray