Consider the Diophantine equation $x^3+x+y^3+y = z^3 + z$ for positive integer $x,y,z$.
I tried small values and got some near equalities : $(5,6,7)$ and $(12,16,18)$ are true up to value $2$. $( 5^3 + 5 + 6^3 + 6 = 7^3 + 7 + 2 )$.
$(6,8,9)$ is true up to value $4$. And it appears that there are infinite near misses having a value of $80$ or $12$ or a power of $2$.
I already tried to find a contradiction mod $30$ and mod $7$ but did not find any.
I noticed that $x^3 + x$ can be squarefree and I cannot find infinite descent.
Does the polynomial ($x^3+x+y^3+y -(z^3 + z)$) factor perhaps ? It is irreducible in $Z$ though.
I must note that I can solve the similar looking $x^3 -x + y^3 -y = z^3 - z$ over the integers which is also irreducible over $Z$. (for instance by letting $z = u+v$ and $x = u - v$ reducing a subset of the solutions to a Pell equation )
Maybe Im close. I would like to solve this Diophantine without using p-adic.
And I also wonder about $x^3 + x + y^3 + y - z^3 - z = (12,80,2^n)$ and $x^5 + x + y^5 + y = z^5 + z$ for $x,y,z>0$.