The Most Outrageous Conjecture in the World

Donald Knuth, Enrico Bombieri, and Julia Robinson. Ten years ago I asked a question on MathOverflow about The most outrageous (or ridiculous) conjectures in mathematics. It led to a nice list of many outrageous conjectures. Here I would like to present what I regard as a serious contender for the most outrageous (ridiculous) conjecture. It is originated from the following MathOverflow quetion I asked in 2013 Infinitely many primes, and Mobius randomness in sparse sets, and it is closely related to some aspects of polymath4 (2009). The most outragous conjecture (informal form): There is an inherent difficulty to prove infinitude of primes, and related number theoretic results (like Mobius randomness) for explicit, sparse sets of integers. The terms “sparse,” “explicit,” and “inherent difficulty” are negotiable and we will give them the most outrageous interpretations. By “sparse sets” we mean sets of integers that has less than elements smaller than , for every . For example, (i)…