Number of integral solutions to elliptic curve $\binom n2=\binom m3$.

Question

I am wondering if there are infinitely many integral solutions to the equation: $$ {n \choose 2} = {m \choose 3}. $$ Also, do the solutions have a general form? From what I know, this is an elliptic curve, but I don't have much knowledge about elliptic curves. If this problem is solvable using standard methods, I would also like to have a reference for learning about elliptic curves.

Answer 1

Clearing the denominators and multiplying everything by $3^3=27$ yields the equivalent equation $$81n^2-81n=27m^3-81m^2+54m.\tag{1}$$ Now by setting $y:=9n-5$ and $x:=3m-3$ we get the minimal Weierstrass equation $$y^2+y=x^3-9x+20,\tag{2}$$ which defines an elliptic curve because its discriminant is nonzero. Note that if $(m,n)$ is an integral solution to $(1)$, then $(x,y):=(3m-3,9n-5)$ is an integral solution to $(2)$. By Siegel's theorem an elliptic curve with rational coefficients has only finitely many integral points. The integral points of $(2)$ are $$(−3,4),\ (-2,5),\ (0,4),\ (1,3),\ (3,4),\ (6,13),\ (10,30),\ (12,40),\ (27,139),\ (63,499),\ (105,1075).$$ These correspond to the following solutions $(m,n)$ to $(1)$: $$(0,1),\ (\tfrac13,\tfrac{10}{9}),\ (1,1),\ (\tfrac43,\tfrac89),\ (2,1),\ (3,2),\ (\tfrac{13}{3},\tfrac{35}{9}),\ (5,5),\ (10,16),\ (22,56),\ (36,120).$$

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In general, finding all integral points on an elliptic curve is not an easy task. There are computer packages that can do this for you, if the coefficients of the Weierstrass equation are not too large. There is also the $L$-functions and Modular Forms Database, which is where I found the integral points of this particular curve; see this page.