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I consider the following equation with conditions of obtaining solutions

$$a^m+nx^2 = y^n$$

This equation has solution when $a$ is an even prime and $x, y, m$ are positive integers with $(nx, y) = 1$ and $n>1$.

If we fix $a$ as an odd prime and without restriction on $x, y, m$ and $n$ (may be odd or even) with $n > 1$ and $(nx, y) =1$, can we have solutions?
If yes, how to determine such solutions?
If there is empty solution, how to conclude?

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    For $n=2$, you have a *Pellian equation* $y^2-2x^2=a^m$. If $a\equiv\pm1\pmod8$ then $y^2-2x^2=a$ has infinitely many solutions, as does $y^2-2x^2=a^m$ for each $m$. The easiest way to see this is via the ring of integers in the field ${\bf Q}(\sqrt2)$.2012-09-28
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    @Gerry Myerson! if n= 3 and n = 5, what are the solutions? How can we obtain? Thank you for the reduced pellian equation.2012-09-28
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    How much do you know about *elliptic curves*?2012-09-28
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    @Gerry Myerson! To be frank, I am an average in elliptic curves. Still, I will try to understand, if you answer my question with the elliptic curves background.2012-09-28
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    Well, when $n=3$ you've got $a^m+3x^2=y^3$, which is an elliptic curve. As you may know, it's very hard to say anything general on whether such equations have integer solutions or not. It's even worse when $n=5$ --- that's a hyperelliptic curve, about which much less is known.2012-09-28
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    @Gerry Myerson! because of your reply, I got more interest to know more about my equation at n = 3 and n = 5. Please explain the solving solutions in the case of n = 3 of elliptic curve, even there is no integer solutions. Kindly explain with graph (if possible).2012-09-29
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    I'm sorry, but explaining elliptic curves takes a lot more than a comment and a graph. Get yourself a book like Silverman's and start reading. But first make sure you know your elementary Number Theory, your Algebraic Geometry, your Commutative Algebra, and your Complex Analysis, as you will need them.2012-09-29
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    @Gerry Myerson! or show some steps at least. So that, I will do the remaining part. Do the case for n =3 and I will try for n= 5 or other cases. I am good at elementary number theory, algebra and Real and complex analysis. I am bit poor in Algebraic Geometry. Kindly do some steps for n = 3.2012-09-29
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    @GerryMyerson! you only can help me in this aspect. Please solve...2012-10-01
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    For $n=3$, you can rewrite the equation as $X^2=Y^3-27a^m$ (where $X=9x$ and $Y=3y$). This is called a *Mordell equation*. There is no general way to tell how many integer solutions it has for given $a$ and $m$, or even whether it has any at all. There is an enormous amount of literature about it, which I encourage you to search.2012-10-01
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    @GerryMyerson! Thank you so much for giving such a great help. I will work on this. Thank you.....2012-10-02

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