# Solutions of $y^2=\binom{x}{0}+2\binom{x}{1}+4\binom{x}{2}+8\binom{x}{3}$ for positive integers $x$ and $y$

I was interested in create and solve a Diophantine equation similar than was proposed in the section D3 of [1]. I would like to know what theorems or techniques can be applied to prove or refute that the Diophantine equation of the title has a finite number of solutions, I don't have the intuition to know it. Our equation is given as $$y^2=2^0\binom{x}{0}+2^1\binom{x}{1}+2^2\binom{x}{2}+2^3\binom{x}{3},$$ thus using the definition of binomial coefficients we are interested in to solve this equation for positive integers $$x\geq 0$$ and $$y\geq0$$ $$3y^2=4x^3-6x^2+8x+3.$$

Computational fact. I got up to $$10^4$$ that the only solutions $$(x,y)$$ for positive integers $$x,y\geq 0$$ are $$(x,y)=(0,1)$$,$$(2,3)$$, $$(62,557)$$ and $$(144,1985)$$. For example our third solution is $$3\cdot 557^2=930747=4\cdot(62)^3-6\cdot (62)^2+8\cdot(62)+3.$$

Question. Does the equation $$y^2=\binom{x}{0}+2\binom{x}{1}+4\binom{x}{2}+8\binom{x}{3}$$ have a finite number of solutions for positive integers $$x,y\geq0$$ ? If it is very difficult to solve, what work can be done? Many thanks.

## References:

[1] Richard K. Guy, Unsolved Problems in Number Theory, Problem Books in Mathematics, Unsolved Problems in Intuitive Mathematics Volume I, Springer-Verlag (1994).

• Hi @GerryMyerson many thanks for your comments and attention on my posts, I read these and I appreciate all the comments. If this question result a good post I would like to dedicate this to you. Jul 26, 2019 at 22:20
• Using @ only alerts users who have already engaged with the question. Jul 26, 2019 at 23:53

Start with $$3đŠ^2=4đ„^3â6đ„^2+8đ„+3.$$ The change of variables $$x=X/12$$ and $$y=Y/36$$ gives the equation $$Y^2 = X^3 - 18X^2 + 288X + 1296.$$ Entering this into the LMFDB leads to the page http://www.lmfdb.org/EllipticCurve/Q/315936/g/1 . So your elliptic curve, after another change of variables to get rid of the $$X^2$$ term (replace $$X$$ by $$X+6$$) gives the curve $$Y^2 = X^3 + 180X + 2592. \quad(*)$$ According to the LMFDB, the curve $$(*)$$ has Mordell-Weil rank 2, generated by $$\left(-6, 36\right)$$ and $$\left(18, 108\right)$$. And in the form $$(*)$$, there are the following integral points, where only the point with positive $$Y$$ is listed: $$(-6, 36) ,\; (-3, 45) ,\; (18, 108) ,\; (24, 144) ,\; (28, 172) ,\; (738, 20052) ,\; (1722, 71460) ,\;(6189, 486891) .$$ This leads to the points $$(0, 1),\; (1/4, 5/4),\; (2, 3),\; (5/2, 4),\; (17/6, 43/9),\; (62, 557),\; (144, 1985),\; (2065/4, 54099/4)$$ on your original curve. giving the 4 integral points that you found.
• Many thanks also for your great answer. I wondered that maybe an interesting problem can be, if it has a good mathematical content, to study the solutions of $P(x)={n\brace m}$, where $P\in\mathbb{Z}[X]$ and ${n\brace m}$ are the Stirling numbers of the second kind. If do you like this problem feel free to study it as a present for you. Aug 28, 2019 at 8:37
The curve $$y^2 = 1+\frac{8}{3} x - 2 x^2 + \frac{4}{3} x^3$$ is elliptic. Siegel's theorem says it has only finitely many integral points.
• Many thanks I'am going to accept the other answer but your is also perfect. I provide you other equations if you want to study these. The first is $y^2=(x)_0+(x)_1+(x)_2+(x)_3$, where $(x)_n$ denotes the Pochhammer symbol. It is the equation $y^2=(x+2)^3-2(x+2)^2+1$, and the second is $y^2={x\brace 0}+{x\brace 1}+{x\brace 2}+{x\brace 3}$, where ${n\brace m}$ denotes Stirling numbers of second kind. Using the MathWolrd article Stirling Number of the Second Kind it should be the equation $2y^2=2^x+3^{x-1}+1$. Jul 28, 2019 at 18:49
• Few minutes ago I wondered about a different problem as a present for you: the relationship of Iannucci's equation ([1]) and the sequence A277172 (see also the author of this sequence) from the OEIS. Exponentiating Iannucci's equation and using Fermat's little theorem one has that the solutions $n$ satisfy the congruence. I think that maybe you're interested in this because you've added comments in this article from OEIS. See Theorem 2 from Iannucci's article. References: [1] Iannucci, On the Equation $\sigma(n)=n+\phi(n)$, Journal of Integer Sequences, Vol. 20 Article 17.6.2 (2017). Sep 9, 2019 at 21:01