Infinitely Large Prime Gaps!
Infinitely Large Prime Gaps! #mathclass #shorts
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We currently cannot prove that there are infinitely many prime gaps of size 2 (also known as "twin primes"), though we certainly think that there are. But it turns out it's trivial to prove that there are infinitely many prime gaps of any arbitrarily large size that you'd like!
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The idea is that you can construct prime gaps guaranteed to be at least so large by using the factorial. Consider something small like 5!. Since 5! is guaranteed to have, as factors, 1, 2, 3, 4, and 5, that means that you can easily construct a string of numbers that also have those factors. 5! + 2, for example, must be an even number, since 5! is even. 5! + 3 must be a multiple of 3 since 5! was a multiple of 3. The same holds for 5! + 4 and 5! + 5, thereby guaranteeing at least four numbers in a row that are all composite.
Even beyond that, any multiple of 5! (so 2(5!) and 3(5!) and so on) are guaranteed to have the same property!
#mathclass #shorts #primegaps
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