For integers $x$, $y$ and $z$, if $x^2+y^2=z^2$ then the ordered triple $(x,y,z)$ is called a Pythagorean triple. It is well known that Pythagorean triples $(x,y,z)$ with $2\mid y$ have the form $(k(m^2-n^2),2kmn,k(m^2+n^2))$ with $k,m,n\in\mathbb Z$.
Ramanujan's tau function defined over $\mathbb Z^+=\{1,2,3,\ldots\}$ is given by $$q\prod_{n=1}^\infty (1-q^n)^{24}=\sum_{n=1}^\infty\tau(n)q^n\quad \ (|q|<1).$$ It plays an important role in the theory of modular forms.
Here I propose a conjecture connecting the tau function with Pythagorean triples.
Conjecture. There are no Pythagorean triples of the form $(\tau(k),\tau(m),\tau(n))$ with $k,m,n\in\mathbb Z^+$. In other words, the equation $x^2+y^2=z^2$ has no solution over the range of the tau function.
This conjecture implies Lehmer's conjecture which asserts that $\tau(n)\not=0$ for all $n\in\mathbb Z^+$.
I have checked the Conjecture via Mathematica and found no counterexamples.
QUESTION. Can one provide a concrete counter-example to the above conjecture?
Your comments are welcome!
