I'm reading The Music of the Primes by Marcus Du Sautoy and I came across a page with the following excerpt about Leonhard Euler:
"The role of the court mathematician is perfectly illustrated by a story that was told of Euler's time in St. Petersburg. Catherine the Great was hosting the famous French philosopher and athiest Denis Diderot. Diderot was always very damning of mathematics, declaring that it added nothing to experience and served only to draw a veil between human beings and nature. Catherine, though, quickly tired of her guests...Euler was promptly called to her court to assist in silencing the insufferable athiest. In appreciation of her patronage, Euler duly consented and addressed Diderot in serious tones before the assembled court. 'Sir, $(a+b^n)/n=x$, hence God exists; reply'. Diderot is reported to have retreated in the light of such a mathematical onslaught."
Nothing more is written about this and the book quickly switches to explain Euler's genius through his solution to the Bridges of Konigsberg. I am still left to wonder though what was meant by the above passage with the equation
$$\frac{a+b^n}{n}=x$$
Why did this "defeat" Diderot and prove god exists?
The genius of Euler's "claim" lies in its facile stupidity, such that even the mathematically naive Diderot would immediately recognize it as garbage. Diderot would have been expecting an erudite argument containing a subtle logical flaw, which he could have handled well. Blatant nonsense coming from the mouth of the world's most eminent mathematician would have wrong-footed Diderot. How could Diderot, as an admitted non-mathematician, accuse such an expert of making a "mathematical" claim that is so obviously stupid that one would hardly know where to begin to refute it? Euler must have recognized Diderot as somewhat lacking a sense of humour. Treating Euler's claim as a joke would have defused it completely.
Edit: Well, I should have followed @Thomas Andrews' advice to check the authenticity of the story first. Assuming that it is apocryphal, the above analysis should be taken merely as an account of why the story is credible and "has legs".