Question

What integer solutions are there to `1/(x+y)+1/(x-y) = 4/9` ?

Answer 1

Integer solutions are: `y=+-3` (with `x=6`) or `y=+-10` (with `x=-8`).

Explanation:

Suppose:

`1/m + 1/n = 4/9" "` for some integers `m, n`

Dealing first with the case `m, n > 0`:

Without loss of generality `m <= n` and we have:

`2/9 <= 1/m < 4/9`

Taking reciprocals and reversing the inequalities, that means:

`9/4 < m <= 9/2`

Hence `m = 3` or `m = 4`

If `m=3` then:

`1/n = 4/9 - 1/3 = 1/9" "` which works.

If `m=4` then:

`1/n = 4/9 - 1/4 = 7/36" "` which does not work.

So the only positive `m, n` with `1/m+1/n = 4/9` are `3, 9`

Hence we can put `x+y=9` and `x-y=3`, finding `x=6`, `y=3` and:

`1/(x+y)+1/(x-y) = 4/9`

Another solution is given by `x=6` and `y=-3`.

What about negative values of `n` ?

If `1/m > 4/9` then `m=1` or `m=2`

If `m=1` then `1/n = 4/9-1 = -5/9" "` which does not work.

If `m=2` then `1/n = 4/9 - 1/2 = -1/18" "` giving `n=-18`

Hence we can put `x = -8` and `y = +-10` to get two further integer solutions.