Question

Beginners question on Number Theory: if I have an expression of the form:\[p^2(1-\frac{a}{r})^m+q^2(1-\frac{b}{r})^m\]where a, b, p, q, r and m are all positive integers and both a and b are less than r, then is it valid to argue that the expression can never generate an integer?

Answer 1

by "beginners question" I meant that I am the beginner here, not necessarily that the question is simple. :)

Answer 2

Do we know whether all the unknowns are distinct? Other than \(a,b\) and \(r\), I mean.

Answer 3

yes - they are all distinct

Answer 4

\[p^2(1-\frac{a}{r})^m+q^2(1-\frac{b}{r})^m\\=\left(\frac{p^2}{r^m}\right)(r-a)^m+\left(\frac{q^2}{r^m}\right)(r-b)^m\]
clearly, if m is an even number, \(m=2k\)
\[\frac{p^2}{r^{2k}}=\left(\frac{p}{r^k}\right)^2\]
p may be chosen as a multiple of r^k, thus an integer can be obtained

Answer 5

thanks @sirm3d