When and who was the first person to discover a correct formula for the real number $r$ in terms of any given three positive distinct integers $x<y<z$ such that $$x^r + y^r = z^r\,.$$
The formula: let $$t=\log(y/x) / \log(z/x)$$ and define $f(t)$ as function of real variable $t$ and positive integer $n$ when $n$ tends to infinity:
$$f(t) = \frac{3t}{2} + (2t-1)\left(\frac{4t}{3} - 1\right) + \left(\frac{5t}{2} - 1\right)\left(\frac{5t}{3} - 1\right)\left(\frac{5t}{4} - 1\right) + \cdots + \left(\frac{nt}{2} - 1\right)\left(\frac{nt}{3} -1 \right)...\left(\frac{nt}{n-1} -1 \right)$$
Where $t$ is positive real number less than one, then: $$(x/z)^r = (\log(z/y) / \log(z/x))\cdot f(t)\,,$$or simply $r = \log \left((1-t)f(t)\right) / \log(x/z)$.
So $r$ is therefore obtainable as a function of the three positive distinct given integers $x<y<z$.