Amongst philosophers, Socrates is an example of a genius with a great influence on human history who never wrote anything. Almost all facts which are known about his revolutionary ideas are written by his students like Plato and Xenophon.
Question: What are examples of genius mathematicians in modern or ancient eras whose ideas had a great influence on development of mathematics but never wrote or published anything?
There are influential posthumous papers by authors who did not publish them.
Galois had an immense influence on algebra because of the publication of something he wrote the night before he died in a duel. He gave us the word "group".
Thomas Bayes had an influential posthumous paper in which he found the conditional probability distribution of a random variable $P$ whose marginal distribution is uniform on $[0,1]$, given the observation of the number of successes in $n$ trials that are conditionally independent given $P$, where the conditional probability of success on each trial, given $P$, is $P$. He did publish things while he lived.
Mary Cartwright never published her proof of the irrationality of $\pi$. A question she set on an examination was to fill in its details. We know of it only because Sir Harold Jeffries included it in an appendix in one edition (and not in other editions) of one of his books, and he leaves the impression that he knew of it only because of its occurrence on that examination.
The words "never" and "anything" are somewhat restrictive (and we know that absolute statements are always wrong). Nowadays, you can't be recognized at all when you're not publishing. Additionally, the term "Genius" is hard to define...
One person came to my mind is Grigori Yakovlevich Perelman. He hardly published anything, did not even defend his dissertation, and was one of the few leading researchers at his institute who only was a "PhD candidate". Of course, strictly speaking, he did publish something, particularly his proof of a generalization of the Poincaré conjecture. But this was not published in some mathematical Journal, but only on arXiv.
I think that someone who casually finds a proof for a generalization of a conjecture that dozens of mathematicians had been working on for 98 years, and doesn't give a ... care about the formal process of scientific publications qualifies as a genius, and is close enough to "never publishing anything" to be mentioned here, at least...
Riemann published but 15 papers in his lifetime. One of my professors once told me, perhaps apocryphally, that one of those papers contained an error, and that his distress over the error led him to poor health and eventually his death.
After his death, his housekeeper "cleaned up" his papers, losing who knows how many of his mathematical ideas forever.
Pierre de Fermat-Some of his works and theorems were published by his son. His works, including Fermat's last theorem $(a^n+b^n\neq c^n \,\forall n>3,a,b,c,n\in\mathbb{Z})$ were not published until his death.
I think Thales of Miletus is the most obvious answer, although perhaps a bit boring. From wiki:
Thales involved himself in many activities, taking the role of an innovator. Some say that he left no writings, others say that he wrote On the Solstice and On the Equinox. (No writing attributed to him has survived.)
He is said to have basically invented mathematics, or rather the axiomatic proof-based approach to mathematics.
Joshua King could be an example of a genius mathematician who never gave a lecture and published almost no papers but one.
To echo others, the question leaves a lot open to interpretation; however, my vote would have to go to Gosset, the inventor of the "t Distribution."
William Sealy Gosset (13 June 1876 – 16 October 1937) was an English statistician. He published under the pen name Student, and developed the Student's t-distribution. (wiki)
Given that Gosset published under a pen name, it could be argued that he never "published" .
Many people are familiar with the name "Student" but not with the name Gosset. In fact Gosset wrote under the name "Student" which explains why his name may be less well known than his important results in statistics. He invented the t-test to handle small samples for quality control in brewing. Gosset discovered the form of the t distribution by a combination of mathematical and empirical work with random numbers, an early application of the Monte-Carlo method. (http://www-history.mcs.st-andrews.ac.uk/Biographies/Gosset.html)
Pythagoras and other members of his school whose names haven't survived seem like an obvious choice (along with Thales, given in another answer). Surely they had a huge effect on mathematics, which is similar to the influence of Socrates on Philosophy, but we have very little extant writings from Pythagoreans and none from Pythagoras himself.
Honestly $\textbf{Gregori Perelman}$ is a genius mathematician for me the guy literally who's gave up 1 million US dollars and now living with his mom in Russia.
He said, ""I'm not interested in money or frame, I don't like to display an animal in a zoo. I'm not a hero of mathematics. I'm not even that Succsefull; that is why I don't want to have everybody's looks at me"". This guy seriously a legend of mathematics.
For me, one million dollars really nothing even you understand math at level. Honestly I respects Dr Perelman that so much. Dude did so much for the contribution of academic world and wanted nothing in return but to live his own life out of the public eye.
