How many solutions does this equation have?

While studying a particular sequence of numbers, I came across the following equation. I am looking to find, in terms of two positive integer constants $a$ and $b$ , the number of unique positive integer solutions, $(x,y)$ , to the following equation:

$x^2+ax=y^2+by$

Any insight would be much appreciated, and if necessary I can give further context for the question.

Thanks!

#NumberTheory

Easy Math Editor

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Comments

I don't know about the general case, where x and y can go from negative infinity to positive infinity. But here's an algorithm for determining the number of (x,y) solution pairs for x between two finite integer numbers (p,q).

0) Constants (a,b) are given as inputs. (p,q) is a predetermined range for x.
1) Evaluate all integer x values in range (p,q)
2) For each x, substitute into the left side and solve the resulting quadratic for y
3) Store (x,y) pairs corresponding to real integer y solutions.

Not very insightful, but you said "any insight"

Steven Chase - 2 years, 11 months ago

Hint: Multiply both sides by 4, then complete the square.

Hint 2: Rearrange them to get $(2x+a)^2 - (2y + b)^2 = a^2-b^2$ . Factorize both sides.

Hint 3: Divide both sides by $(a-b)(a+b)$ .

Hint 4: $1 = 1\times1=-1\times-1$ .

Pi Han Goh - 2 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$