Determine all possible pairs (p, q) of
prime numbers that satisfy this set of simultaneous equations:
where each of a and b is a positive integer.
Validate your answer by providing sufficient analytic reasoning.
(p,q) = (3,2) is the only solution.
From the first equation, we see that p > q, otherwise a would not be a positive integer.
From the second equation, pq - q = (p-1)q = b^2.
There are then two cases to consider:
1) (p-1) = q = b. Since p and q are both primes, 3,2 is the only possible solution as the only pair of consecutive primes.
2) (p-1) = q^n, where n is an odd number greater than 2. From this, p and q must have opposite parity, and since p > q, q must equal 2. Any solutions must then have q = 2, p = 2^n + 1, where n is odd. However, numbers of the form 2^n + 1 can only be primes if n itself is a power of 2 (and therefore even), so there are no solutions for this case.
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Posted by H M
on 2023-05-31 11:47:40 |
Solution
