Got an argument and numerical support againts the $abcd$ conjecture with extra gcd conditions (observe that this is different from the $abc$ and the $abcd$ conjectures).

This thesis p. 20 defines the $abcd$ conjecture with extra gcd conditions.

$a,b,c,d$ are pairwise coprime integers, $a+b+c+d=0$ and $d$ is positive with largest absolute value.

$$ \lim\sup \frac{\log(d)}{\log\left({\rm rad}(abcd)\right)}= 1$$

Couldn't find reference in the thesis for this claim.

Q1 What is a reference for this conjecture?

The argument against it. From an identity for sum of two fourth powers define:

```
a(t)=3*t^13 - 38*t^12 + 516*t^11 - 4240*t^10 + 20208*t^9 - 63744*t^8 + 144960*t^7 - 248192*t^6 + 325632*t^5 - 328704*t^4 + 251904*t^3 - 139264*t^2 + 49152*t - 8192
b(t)=t^13 - 40*t^12 + 188*t^11 + 472*t^10 - 9456*t^9 + 50496*t^8 - 155072*t^7 + 316160*t^6 - 454656*t^5 + 473088*t^4 - 354304*t^3 + 182272*t^2 - 57344*t + 8192
c(t)=t^13 + 14*t^12 - 460*t^11 + 3664*t^10 - 17616*t^9 + 62208*t^8 - 166976*t^7 + 334208*t^6 - 488448*t^5 + 513024*t^4 - 378880*t^3 + 188416*t^2 - 57344*t + 8192
d(t)=3*t^13 - 40*t^12 + 540*t^11 - 3944*t^10 + 16368*t^9 - 40512*t^8 + 55488*t^7 - 15616*t^6 - 92160*t^5 + 196608*t^4 - 208896*t^3 + 133120*t^2 - 49152*t + 8192
```

They are pairwise coprime and satisfy

$$ a(t)^4+b(t)^4-c(t)^4-d(t)^4=0$$

By the degree argument, for fixed $t$ whenever they are coprime the abcd quadruple is of good quality. $\deg (d(t)^4) = \deg (a(t)b(t)c(t)d(t))=52$.

The for integer $t$, the $\gcd$ of any of them is bounded by the resultant. The radical of the product of all resultants is $6$. Working modulo $6$, they are pairwise coprime when $t \equiv 5 \pmod{6}$.

To reduce the radical sufficiently, chose natural $u$ coprime to $6$, natural $n$ sufficiently large and solve $a(r) \equiv 0 \pmod{u^n}$. Then $a(r)=O(r^{13})$ and upper bound for the radical is $6 u O(r^{12})$, which for sufficiently large $n$ gives quality of $\frac{52}{51+o(1)}$.

To ensure coprimality with the chinese remainder theorem set $t \equiv 5 \pmod{6},t \equiv r \pmod{u^n}$.

This makes the quality $\frac{52}{51+o(1)}$ and the constant probably can be made explicit.

Experimentally, in very short time found $1520$ quadruples of quality at least $1.019$ ($\frac{52}{51}=1.0196\ldots$). To avoid factorization, gave upper bound for the radical. Some of them were with very large radical and large merit (not sure the merit makes sense in this case).

Q2 Is this really a counterexample to the conjecture?