Conjectures disproven through counterexample are sometimes referred to as "false conjectures" (cf. This functoriality conjecture implies all the other conjectures presented so far. The theory consists both of theorems and many conjectures and open questions. Euler developed some of Fermat's ideas and disproved some of his conjectures. Still, philosophical perspectives, conjectures, and presuppositions, often overlooked, remain necessary in natural science. Popper dedicated his "Conjectures and Refutations" to Hayek. Goldbach and Riemann's conjectures remain unproven and unrefuted. The Riemann hypothesis is one of the most important conjectures in mathematics. The modularity theorem is a special case of more general conjectures due to Robert Langlands. This is a list of mathematical conjectures. Many conjectures revolving about primes have been posed. This monster vertex algebra was also used to prove the monstrous moonshine conjectures. He made a series of conjectures on class field theory. Most cases of these extended conjectures have not yet been proved. Modern scholarship suggests that these conjectures are not correct.