Consider the identity $$ (x+z)^5+(y-z)^5 = (-3 x + 4 y)^2 (x + y)^3 + (x+y) f(x,y,z) $$

Where $f(x,y,z)=(-8*x^4 + 5*x^3*y + 24*x^2*y^2 - 9*x*y^3 - 15*y^4 + 5*x^3*z - 5*x^2*y*z + 5*x*y^2*z - 5*y^3*z + 10*x^2*z^2 - 10*x*y*z^2 + 10*y^2*z^2 + 10*x*z^3 - 10*y*z^3 + 5*z^4)$.

The curve $C : f(x,y,z)=0 $ is genus 1, have infinitely many rational and integral points since it is projective.

For a point $(x,y,z)$ on $C$, the identity becomes.

$$ (x+z)^5+(y-z)^5 = (-3 x + 4 y)^2 (x + y)^3 \qquad (1) $$

(1) has infinitely many integer solutions.

Sum of two coprime fifth powers being squarefull infinitely often contradicts the $abc$ conjecture, so $abc$ implies $\gcd(x+z,y-z) > 1$ (actually it implies the $\gcd$ is sufficiently large, since clearing a small gcd will produce abc triples of sufficiently good quality).

$C$ is birationally equivalent to $ E: y^2 = x^3 - \frac{57648010}{243}x - \frac{346032180025}{19683}$ and computing solutions to (1) gives large gcd, as implied by $abc$. Computing the gcd of the symbolic maps from the Weiersstras model give $\gcd=7$ (modulo errors).

Other similar identities exist, including genus 0 curves. The parametrization of genus 0 and $abc$ for polynomials implies common factor, though the genus 1 case is not clear to me.

Why the $\gcd$ is sufficiently large?

Is there an unconditional proof that for all similar identies the $\gcd$ will be sufficiently large? ($abc$ implies this).

**Added** Charles asked in a comment about points of infinite order on $C$.

Here are some:

```
(27, 1, 15)
(-1343, -1184, 279)
(-113217, -61531, 74507)
(-1038297, 1267624, 243888)
(18490353467, 11046438881, 1513527591)
(17139398481243, 15697885061884, 4151488981525)
(-26723833000980177, 15287849768762549, 47286394561187571)
(-1316887777770612905003, -1407701177079680302604, 837630236024655513348)
```

If $(-u,v)$ **note the minus** is on $E$ a map to (x,y,1) on $C$ is:

x,y=(-170318769169205125-1417766490*u^3-531441*u^4+1975193766630*u^2-40040866545750*v+99059869755*v*u+40507614*v*u^2)/(-601680754627470*u-63038098935*u^2+531441*u^4+1748578671*u^3+135424192071124075), (-2125764*u^4-4017005055*u^3+4664819321190*u^2-261600328098900*u-279987440048888425-91293175724310*v+243146953035*v*u+60761421*v*u^2)/(-1805042263882410*u-189114296805*u^2+1594323*u^4+5245736013*u^3+406272576213372225)