Each of p and q is a
prime number and each of x and y is a
positive integer, with x greater than 1.
Find quadruplet(s) (p, q, x, y) that satisfy this equation:
px - qx = 2y
providing adequate reasons as to why there are no further solutions.
*** Adapted from a problem which appeared at the Brazilian Mathematical Olympiad in 1997.
Both primes are odd.
If x is odd, LHS can be factored:
(p-q)*(p^x-1 + qp^x-2 + . . . + q^x-1)
Each term of the second factor is odd, and there is an odd number of them so that their sum is odd and thus cannot divide a power of two.
So there are no solutions when x is odd.
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Posted by xdog
on 2022-06-06 12:04:36 |
when x is odd
