There could be several personal, social, philosophical and even political reasons to keep a mathematical discovery as a secret.
For example it is completely expected that if some mathematician find a proof of $P=NP$, he is not allowed by the government to publish it as same as a usual theorem in a well-known public journal because of high importance and possible uses of this special proof in breaking security codes which gives an undeniable upper hand to the state intelligence services with respect to other countries. Also by some social reasons publishing such a proof publicly is not suitable because many hackers and companies may use it to access confidential information which could make a total chaos in the community and economy.
The example shows that in principle it is possible to have some very significant brilliant mathematical proofs by some genius mathematicians which we are not even aware of. But in some cases these "secrets" unfold by an accident or just because they lost their importance when some time passed and the situation changed.
Question: What are examples of mathematical discoveries which were kept as a secret when they discovered and then became unfolded after a while by any reasons?
Best Answer
Niccolò “Tartaglia” Fontana invented the first general method to find the roots of an arbitrary cubic equation (based on earlier work by himself and others on how to solve cubics of particular forms), but kept his method secret so as to preserve his advantage in problem-solving competitions with other mathematicians.
He divulged the secret to his student Gerolamo Cardano on condition that Cardano keep the secret, and there was a bitter dispute between the two when Cardano broke his promise.