Today, I heard of something so called Goldbach's conjecture from my mathematics teacher in the class. This was one of the most interesting things that I have ever heard in mathematics.
This made me curious to study a bit more about conjectures. The definition of conjecture on google says that:
A conjecture is an opinion or conclusion formed on the basis of incomplete information.
Now the question which is stuck in my mind is: What is the use of conjectures in modern mathematics?" Are they used in problem solving as we use theorems/lemma?
If it is the case then how can we use something to solve a problem which is not even known with certainty (Which we can't prove)??
Best Answer
From wikipedia:
Sometimes a conjecture is called a hypothesis when it is used frequently and repeatedly as an assumption in proofs of other results. For example, the Riemann hypothesis is a conjecture from number theory that (amongst other things) makes predictions about the distribution of prime numbers. Few number theorists doubt that the Riemann hypothesis is true. In anticipation of its eventual proof, some have proceeded to develop further proofs which are contingent on the truth of this conjecture. These are called conditional proofs: the conjectures assumed appear in the hypotheses of the theorem, for the time being.
These "proofs", however, would fall apart if it turned out that the hypothesis was false, so there is considerable interest in verifying the truth or falsity of conjectures of this type. Instead, a conjecture is considered proven only when it has been shown that it is logically impossible for it to be false.
But on the other hand, not every conjecture ends up being proven true or false. The continuum hypothesis, which tries to ascertain the relative cardinality of certain infinite sets, was eventually shown to be undecidable (or independent) from the generally accepted set of axioms of set theory. It is therefore possible to adopt this statement, or its negation, as a new axiom in a consistent manner (much as we can take Euclid's parallel postulate as either true or false).