Question

Twice the square of the first subtracted from the square of the second is -167, what are the two integers?

Answer 1

Even if we assume the integers are both positive, there are an infinite number of solutions to this question. The minimal (positive) values are
`(11,12)`

Explanation:

If the first integer is `x` and the second integer is `y`

`y^2-2x^2 = -167`

`y^2 = 2x^2-167`

`y = +-sqrt(2x^2-167)`
`color(white)("XXXX")`(from here on, I will limit my answer to positive values)

if `y` is an integer
`rArr 2x^2-167 = k^2` for some integer `k`

We could limit our search by noting that `k` must be odd.

Since `x` is an integer
`color(white)("XXXX")` `(k^2-167)/2` must also be an integer

Unfortunately there are lots of solutions for `k` that satisfy the stated conditions:

#{:(k,,first,second), (11,,12,11), (15,,14,15), (81,,58,81), (101,,72,101), (475,,336,475), (591,,418,591) :}#
are the values that I found for `k<1000`
and all of these satisfy the given conditions.

(...and, yes, I know `k=y`).