# Positive square rationals equidistant to 1

Can there be two square rational numbers that are equidistant to $$1$$ i.e. there is a rational number $$a \in [0,1]$$ such that both $$1-a$$ and $$1+a$$ are square rationals?

## 2 Answers

Note that two numbers that sum to 2 are equidistant to 1. Now take your favorite Pythagorean triple

$$x^2 + y^2 = z^2$$

and see that

$$\frac{(x - y)^2}{z^2} + \frac{(x + y)^2}{z^2} = 2$$

Hint : $$1+49 = 2 \times 25$$

• So $a=1/25$ and you assumed that it has numerator $1$ to check if this would work? Thanks though. I wonder if this is related to a more difficult diophantine equation. – quantum Jun 26 at 23:14
• Note: also $1+1681=2\times841$ – J. W. Tanner Jun 26 at 23:16
• @J.W.Tanner there are many many more including $169+8281 = 529+7921 = 1225+7225 = 2209+6241 = 2 \times 4225$ – Henry Jun 26 at 23:21
• @quantum - not quite: but $a=\frac{24}{25}$ works – Henry Jun 26 at 23:22