Reducing a binary quadratic Diophantine example

[JSH]

Decided would be worth it to reduce a binary quadratic Diophantine to demonstrate my method.

Used:

x^2 + y^2 = xy + x + y + 102

And reduced to:  9(-(x+y)+2)^2 + 3s^2 = 9*412

From which I could find more than one solution where one is x = -10 and y = - 8.

To see more detail, you can check out my blog post on Some Math:

http://somemath.blogspot.com/2016/08/reducing-quadratic-diophantine-to-find.html

[Haydon Berrow]

Attached is a short note on why the equation can be solved for N=102 but not for N=100,101 or 1000. This is a second draft, I made an error in sign that propagated but didn't affect the reasoning and conclusion, just the values of x and y in a solution.

[Saffron Waldon]

On Wednesday, November 7, 2018 at 3:11:06 PM UTC, Haydon Berrow wrote:

Attached is a short note on why the equation can be solved for N=102 but not for N=100,101 or 1000. This is a second draft, I made an error in sign that propagated but didn't affect the reasoning and conclusion, just the values of x and y in a solution.

Is there an integer solution for every valid value of N? If y were odd then (y-2x-1) would also be odd and there might be non-integer solutions for x.