There is not "the" conjecture: there are several.
The oldest and probably best known unsolved conjecture in number theory is the Goldbach conjecture. According to it every even integer greater than two can be expressed as the sum of two prime numbers.
Because mathematics is a axiomatic system so that every new statement remains a conjecture until it is proved.
Usually not. If you do use conjectures, you should make it quite clear that the proof stands and falls with the truth of the conjecture. That is, if the conjecture happens to be false, then the proof of your statement turns out to be invalid.
By creating a strong inference, you can then put your ideas to the test. After close observation, you can then rule-out any incorrect guesses.
Inductive Reasoning
A conjecture is a statement that is believed to be true, but has yet to be proven. Conjectures can often be disproven by a counter example and are then referred to as false conjectures.
That would most likely be one of the many as yet unproven conjectures that are believed to have proofs.
No. Conjectures are "good" guesses.
Because mathematics is a axiomatic system so that every new statement remains a conjecture until it is proved.
In geometry, deductive rules can be used to prove conjectures.
prove conjectures
Some words that rhyme with "lectures" are textures, conjectures, and ruptures.
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surmises
Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.
inductive
hypotheses or more generally conjectures should be capable of being refuted see: Karl Popper - Conjectures and Refutations
false