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.
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Servaes
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Hyunbok Wi
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2Are you interested in rational points? Or integer points? – hardmath Apr 24 '19 at 05:56
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@hardmath integer points. – Hyunbok Wi Apr 24 '19 at 06:19
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See https://math.stackexchange.com/questions/85442/are-there-surprisingly-identical-binomial-coefficients and the links there. – Gerry Myerson Apr 24 '19 at 07:31
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Have you followed up on this, Quasi? – Gerry Myerson Apr 26 '19 at 04:19
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Are you still here, Quasi? – Gerry Myerson Apr 27 '19 at 07:46
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@Gerry Myerson what do you mean by follow up? – Hyunbok Wi Apr 28 '19 at 00:21
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I mean, did you go to the link and read what was there? Did you go to the links at that link and read them? Did you find anything of relevance to the question you asked here? – Gerry Myerson Apr 28 '19 at 01:55
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So, followed up, Quasi? – Gerry Myerson Apr 29 '19 at 13:01
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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 $(1)$: $$(0,1),\ (\tfrac13,\tfrac{10}{9}),\ (1,1),\ (\tfrac43,\tfrac89),\ (2,1),\ (3,2),\ (\tfrac{13}{3},\tfrac{35}{9}),\ (5,5),\ (10,16),\ (22,56),\ (36,120).$$
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.
Servaes
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Thanks, can you recommend me a good book/reference about elliptic curves? I only have a background in some abstract algebra and I'm starting to learn Algebraic NT. What would be some other prerequisites? – Hyunbok Wi Apr 24 '19 at 07:34
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@Quasi07 Other prerequisites are some basic algebraic geometry, and before that commutative algebra, which you will also need for algebraic number theory. I can recommend the classic Introduction to Commutative Algebra by Atiyah and Macdonald. – Servaes Apr 24 '19 at 07:47
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2I was introduced to elliptic curves myself by Silverman's book Rational points on Elliptic Curves, which I quite liked. It has a chapter on integral points on elliptic curves as well. – Servaes Apr 24 '19 at 07:50
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