A certain problem in *equal sums of like powers* for $7$th powers entails the elliptic curve,


$$u(u+127^2)(u+129^2) = y^2$$


I was looking at the general case, 


$$u \big(u + (n - 1)^2\big) \big(u + (n + 1)^2\big) = y\_1^2\tag1$$ 


and using this [online *Magma*](http://magma.maths.usyd.edu.au/calc/) to test integer $n>1$ such that it has positive rank. It starts as,


$$n = 6, 13, 16, 18, 22, 23, 32, 33, 35, 36, 37, 41, 42, 43, 44, 45, 46, 50,\dots$$


Checking the OEIS, it turns out it ***may*** be sequence [A228380](http://oeis.org/A228380) but involves the $n$ of the elliptic curve,


$$v(v - 1)(v - n^2) = y\_2^2\tag2$$


such that it has positive rank.



> 
> **Q:** Is it true that the sequence of $n>1$ for $(1)$ and $(2)$ are in fact the same? If so, why?
> 
> 
> 



On the elliptic curves $u \big(u + (n - 1)^2\big) \big(u + (n + 1)^2\big) = y_1^2$ and $v(v - 1)(v - n^2) = y_2^2$
Let $E\_n$ and $F\_n$ be the elliptic curves
$$\begin{aligned}
E\_n : y\_1^2 &= u\bigl(u+(n-1)^2\bigr)\bigl(u+(n+1)^2\bigr), \\
F\_n : y\_2^2 &= v(v-1)(v-n^2).
\end{aligned}
$$
There are only finitely many $n$ values for which $E\_n$ and $F\_n$ are isomorphic, since
$$
\text{numerator of }j(E\_n)-j(F\_n)
=
(n + 3)(3 n + 1)(n^2 - 2 n + 5)(5 n^2 - 2 n + 1)(n^3 - 3 n^2 - n - 1)(n^3 + n^2 + 3 n - 1).
$$
So if the ranks are matching, best guess is that the curves are 2-isogenous. Thus using the formulas in Example 4.5 of *The Arithmetic of Elliptic Curves*, we have
$$ F\_n : y^2=x^3+ax^2+bx\quad\text{with $a=-(n^2+1)$ and $b=n^2$.} $$
The 2-isogenous curve $F\_n/\langle(0,0)\rangle$ is
$$ E : Y^2 = X^3-2aX^2+rX\quad\text{with $r=a^2-4b=(n^2-1)^2$.} $$
Hence the isogenous curve is
$$ E : Y^2=X^3+2(n^2+1)X^2+(n^2-1)^2X = X(X+(n-1)^2)(X+(n+1)^2), $$
which is exactly your curve $E\_n$. In particular, since the curves are $2$-isogenous over $\mathbb Q$, we have $\text{rank }E\_n(\mathbb Q)=\text{rank }F\_n(\mathbb Q)$. This holds for all $n\in\mathbb Q\setminus\{\pm1,0\}$.


The MO question [On the elliptic curve $x(x+a^2)(x+b^2) = y^2$](https://mathoverflow.net/questions/191605/on-the-elliptic-curve-xxa2xb2-y2?rq=1) is related.


