Use Fermat factorization to factor $809009\ldots$

Question

Use Fermat factorization to factor $809009\ldots$

So far I have:

\begin{align} \sqrt{809009} & = 889.449 \\ & = 890 \\[6pt] \sqrt{890^2 - 809009} & = 130\ldots ∉ \mathbb Z \\[6pt] \sqrt{891^2 - 809009} & = 122\ldots ∉ \mathbb Z \\[6pt] & \,\,\,\vdots \\[6pt] \sqrt{899^2 - 809009} & = 28\ldots ∉ \mathbb Z \end{align}

Ideally should equate to a whole number at some point, and according to my professor I shouldn't have to try for more than about 5 values. Not sure if I made an arithmetic mistake or just not using the formula correctly. Any help is appreciated.

Answer 1

$$\sqrt{809009}=899.449$$

Your teacher is right. You can get it in about 5 values.

You should start from 900 rather than 890.

I have written some Python code to avoid doing manual work.

http://www.codeskulptor.org/#user41_l8aZFH9i8fjEuMO.py

$$\sqrt{903^2-809009}=80$$

Hence the factors are $903-80=823$ and $903+80=983$