Using a conjecture to solve a conjecture

Question

Is it mathematically correct to use a conjecture to prove another conjecture ? And if the second one is proved, does that mean that we'll only focus on proving the first ? There are many cases that emerge from this.

This idea came to my mind while studying Goldbach's Conjecture

Answer 1

A mathematical theorem is typically a statement $P\implies Q$ that is logically true under the theory's axioms.

A conjecture is an unproven theorem.

Consider the following valid mathematical proof:
    $P\quad$ and $\quad P\implies (A\implies B);$
    therefore $\quad A\implies B.$
The first line contain the assumptions (also called premises or hypotheses), while the second line contains the conclusion.

• If we know $P$ to be true, and $\big[P\implies (A\implies B)\big]$ to be a theorem, then we can rightly consider $\big[A\implies B\big]$ to be another theorem. In this case, the above proof is a sound argument.

• On the other hand, if we know $P$ to be true, but $\big[P\implies (A\implies B)\big]$ is merely a conjecture, then $\big[A\implies B\big]$ has to be another conjecture.

  1. If the first conjecture becomes proven, then we're back to the previous case, and the second conjecture automatically too becomes proven.
  2. But if the first conjecture becomes disproven, then the above proof, while still valid, becomes unsound, and the second conjecture remains a conjecture. (Even though the second conjecture had been derived from the first, disproving the latter does not automatically disprove the former.)