# 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 . 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:

 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. – user142929 Jul 26 '19 at 22:20
• Using @ only alerts users who have already engaged with the question. – Gerry Myerson Jul 26 '19 at 23:53

Start with $$3𝑦^2=4𝑥^3−6𝑥^2+8𝑥+3.$$ 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. – user142929 Aug 28 '19 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 $2y^2=2^x+3^{x-1}+1$. – user142929 Jul 28 '19 at 18:49
• Few minutes ago I wondered about a different problem as a present for you: the relationship of Iannucci's equation () 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:  Iannucci, On the Equation $\sigma(n)=n+\phi(n)$, Journal of Integer Sequences, Vol. 20 Article 17.6.2 (2017). – user142929 Sep 9 '19 at 21:01