Almost Right-angled Triangles I

Let us call an integer sided triangle with sides $a \leq b \leq c$ barely acute if the sides satisfy $a^{2} + b^{2} = c^{2} + 1$ .

How many barely acute triangles are there with perimeter $\leq 25 000 000$ ?

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.