Help finding all the solutions of the diophantine equation: $ x^2+4(ab)^n=y^2$

Question

I need help finding the solutions of the diophantine equation: $ x^2+4(ab)^n=y^2$. Please be aware the only one that I could find was $ x= \pm (a^n-b^n), y=\pm (a^n+b^n)$ using the quarter squares rule. I was told that are more integral solutions than that. Over $\mathbb{C}$ method was unsuccessfully attempted, not sure where i went wrong. Any help will be greatly appreciated. Thanks.

Answer 1

Factor (as I'm sure you did) like this $$ 4(ab)^n = y^2-x^2 = (y-x)(y+x). $$ As $y-x$ and $y+x$ are of the same parity, they must both be even. Write $a=a_1a_2$ and $b=b_1b_2$ for integers $a_1,a_2,b_1,b_2$. Then $y-x=2(a_1b_1)^n$ and $y+x=2(a_2b_2)^n$, which implies $$ y = (a_2b_2)^n+(a_1b_1)^n, \qquad x=(a_2b_2)^n-(a_1b_1)^n.$$ That should be all of the solutions, I think — your original answer is a special case of this one (in which you did not consider factorizations of $a$ and $b$).

Answer 2

Let $d=(ab)^n$, and let $uv=d$. Set $x+y=2u$ and $x-y=2v$. Then $x=u+v$ and $y=u-v$. This gives all solutions.

Thus our problem is equivalent to finding all pairs $(u,v)$ such that $uv=d$. There are the obvious solutions $u=\pm a^k b^l$, $v=\pm a^{n-k}b^{n-l}$, with $0\le k\le n$, $0\le l\le n$ (the plus signs go together, as do the minus signs). If $a$ and $b$ are distinct primes, these give all the solutions.

In general, suppose that $ab$ has the prime power factorization $$ab=p_1^{e_1}p_2^{e_2}\cdots p_m{e_m}.$$ Then there are
$$2(ne_1+1)(ne_2+1)\cdots(ne_m+1)$$ ordered pairs $(x,y)$ of integers such that $x^2+4(ab)^n =y^2$.