How to find all positive integers $(m,n)$ such that $m^2+527=8n^4$?

How to find all positive integers $$(m,n)$$ such that $$m^2+527=8n^4$$?

I have tried $$n = 3$$ and $$n = 4$$: $$(m,n) = (11,3)$$ or $$(39,4)$$ are the two possible solutions.

However, are there any other solutions and how can I prove that these are the only solutions?

I have tried to perform modulo 3 and 4 on the equation, but in vain.

• Try this: Your equation is equivalent to the equation $$x^2-2y^2=2\cdot527$$ where $x=2n^2$ and $y=m$.
– boaz
Commented Feb 22, 2023 at 10:38
• Thanks @boaz, do you mean $2x^2-y^2=527$? Commented Feb 23, 2023 at 0:26
• No. Note that $(2n^2)^2-2m^2=2\cdot527$ is equivalent to you equation.
– boaz
Commented Feb 23, 2023 at 11:07
• When I simplify your expression, is it $2n^4=527+m^2$ instead? Commented Feb 23, 2023 at 15:11
• I think you mean $(m, n) = (11, 3)$ and $(39, 4)$, not the other way around. Commented Mar 10, 2023 at 17:11

Rewrite the equation in the form $$m^2=8n^4-527$$ and multiply both sides by $$(8n)^2$$ to obtain $$(8mn)^2=(8n^2)^3-4216(8n^2)$$. The integer solutions with positive $$Y$$ for the elliptic equation $$Y^2=X^3-4216X$$ are obtained by SAGE simply by typing in the two lines:

E = EllipticCurve([0,0,0,-4216,0])

E.integral_points()

From the output $$[(0 : 0 : 1), (72 : 264 : 1), (128 : 1248 : 1), (529 : 12075 : 1)]$$ one sees that the solutions for the original equation are obtained from the systems

$$8n^2=72$$, $$8mn=264$$

and

$$8n^2=128$$, $$8mn=1248$$.

Hence the conclusion: $$(m,n)=(3,11)$$ and $$(4,39)$$ are the only solutions in positive integers for $$m^2+527=8n^4$$.

• @Ravi Fernando Of course you're right. Commented Mar 15, 2023 at 9:45
• Thanks a lot for the solution! I wonder if there is any way look for integral points in an elliptic curve (perhaps a manual approach) other than using SageMath? Commented Mar 16, 2023 at 16:07
• @peterkung543 One can manually perform the algorithm described in Cohen, Henri, Number theory. Volume I: Tools and Diophantine equations, Graduate Texts in Mathematics 239. New York, NY: Springer, 2007. Commented Apr 12, 2023 at 17:14
• Thanks a lot! I will try the algorithm in the section 8.7.4. If that's not the right chapter, just let me know. Commented Apr 14, 2023 at 1:59