In this post that I've asked three weeks ago with same title in Mathematics Stack Exchange and identificator **3692235**, for integers $k\geq 1$, we denote the Gregory coefficients as $G_k$. Wikipedia has an article for *Gregory coefficients*, are known as reciprocal logarithmic numbers (I add this as additional reference). I was inspired in problems that I know from the literature (in particular [1], that is from the section of problems of a journal) to solve the following diophantine equation that involves the first few Gregory coefficients in the brackets from RHS
$$y^2=1+\left(\frac{1}{2}n-\frac{1}{12}n^2+\frac{1}{24}n^3\right)$$
where we consider that $y\geq 1$ is integer and $n\geq 1$ also is integer.

Question 1.Prove or refute that the previous diophantine equation $$y^2=1+\sum_{k=1}^3G_k \cdot n^k\tag{1}$$ have no solutions $(n,y)$ when $y\geq 1$ and $n\geq 1$ run over positive integers. Can you find a counterexample?Many thanks.

My claim here was the following, that summarizes the things that I can see here (I don't know if previous question is easy to get). Also I know that $(1)$ is an elliptic curve (but in this post I'm interested in integral solutions).

**Claim.** *Our equation* $(1)$ *can be rewritten as* $n((n-2)n+12)=24(y-1)(y+1)$ *(with help of Wolfram Alpha online calculator). From here we get easily (by contradiction) than* $n$ *is an even integer. And* $n\equiv 0\text{ mod }3$ *or* $n\equiv 2\text{ mod }3$.

I've tested the conjecture stated in previous question for humble sets of integers. On the other hand I'm curious if there is some diophantine equation of the form $y^2=1+\sum_{k=1}^ NG_k n^k$ for some integer $N>3$ for which we can compute at least an integral solution $(n,y)$.

Question 2 (A computational exercise).Can you show an example of diophantine equation $$y^2=1+\sum_{k=1}^N G_k \cdot n^k\tag{2}$$ with at least a solution $(n,y)$, for integers $n,y\geq 1$ as before, where $N>3$?Many thanks.

I tried with my computer the first few values of $N$, the lowest of these integers $N>3$, and for $1\leq n,y\leq 5000$ both integers. If you can to answer **Question 2** with a family of integral solutions, or you can find several examples of $N$ for diophantine equations $(2)$ having solutions feel free to expand your answer of this question.

I don't know if my questions are in the literature. If you know some of these from the literature refer it ansewring the questions as a reference request.

## References:

[1] Fuxiang Yu, *An Old Fermatian Problem: 11203*, Problems, The American Mathematical Monthly, Vol. 114, No. 9 (Nov., 2007), p. 840.

On the Diophantine Equation$1+x+\frac{x^2}{2!}+\ldots+\frac{x^n}{n!}=g(y)$ by Manisha Kulkarni and B. Sury from the book Diophantine Equations edited by N. Saradha, Tata Institute of Fundamental Research, Narosa Publishing House (2008). $\endgroup$Partial sums involving Gregory coefficients that cannot be an integer. I add this as invitation, that in case that these posts have a good/interesting mathematical content for you, you can visit it and provide a contribution as a comment or adding an answer. $\endgroup$