The journey from conjecture (an unproved result, from the Latin conjectura, derived from cum and jacere, literally "to throw together") to theorem can take a very long time: 350 years in the case of "Fermat's Last Theorem," stated by Fermat in the first half of the 17th century and fully proved by the British mathematician Andrew Wiles in 1995. Three and a half centuries of research, sometimes unsuccessful, yet extraordinarily fruitful for number theory through the advances it brought, the new concepts it introduced and the new avenues it opened. What matters to mathematics, then, is not so much a conjecture itself as the search for its proof. Today's stubborn conjectures are the subject of extensive research capable of producing highly fruitful results, even intermediate ones.
Precious conjectures ----------------------
The first open questions we encounter often come from number theory. Yet famous conjectures exist in every branch of mathematics, from geometry and combinatorics to dynamical systems, partial differential equations, set theory, logic and topology.
One of the first unsolved problems that comes to mind in number theory is the Collatz conjecture, or 3n + 1 problem. Proposed by the German mathematician Lothar Collatz around 1937 and later popularized at a conference held at Syracuse University in the United States in the 1950s—hence its name—it is easy to state: choose a strictly positive integer; if it is even, divide it by 2, and if it is odd, multiply it by 3 and add 1; then repeat with the resulting number. Verified for every integer up to 268 (about 2.95 × 10 20 ) following David Barina's recent work (2020), the conjecture states that the process always eventually reaches 1. Starting from 11, for example, we obtain the sequence 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. Starting from 27, it takes 111 steps to reach 1. But a simple statement can sometimes have a complicated solution, and the fact remains that this conjecture still has no complete proof. Progress has come in small steps: in 2003, Ilia Krasikov and Jeffrey Lagarias proved that, for every sufficiently large integer X, the number of integers below X that reach 1 is at least X 0.84. Terence Tao made further compelling and spectacular advances in 2019 and 2020 (see Tangente special issue 76, currently on sale, devoted to Iterative processes, recurrence and recursion).