Flos/Problems/Problem 1

Problem from by

 * Find a rational number such that $5$ added to, or subtracted from, its square, is also a square.

Solution

 * $\dfrac {41} {12}$

Proof
We need to find $3$ squares in arithmetic sequence whose common difference is of the form $5 p^2$.

That is:
 * $\tuple {x^2 - 5 p^2, x^2, x^2 + 5 p^2}$

We can find:

and:

Hence: