How do mathematicians calculate extremely large sets of numbers to prove theorems? Both now and in the past?
A modern example is the Collatz Conjecture, 2\^68 numbers were checked, so what resources do mathematicians use for this? Supercomputers?
And in the past, the Polya Conjecture was disproven by C. Brian Haselgrove in 1956, he calculated more than 2\^361 values, so how was this done when computers weren’t as powerful?
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Note that your question has a fatal misunderstanding. You ask how mathematicians calculate extremely large sets of numbers to *prove* *theorems*, and then give a conjecture as an example.
A conjecture is quite literally an unproven theorem. It means we have yet to logically prove with adequate reasoning that the conjecture is universally and generally true. All we can say is that it appears true and, when we test to see if it’s true, thus far it has always been true. It will stop being a conjecture when we either prove/disprove it, or we stumble upon a counter-example.
The Twin Prime Conjecture is another example.
Real proofs don’t calculate exhaustively because there are, usually, an infinite number of calculations that need to be performed and there will always be this lingering worry that perhaps there exists some example out there to dismantle the conjecture.
Imagine you wanted to prove that the sum of any two odd integers always results in an even integer. You could try checking literally every sum possible, or you could prove it! Look up a proof for this! It’s quite easy to follow and demonstrates perfectly what real mathematics is.
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