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!

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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[KevnReid]

Mind blown. I thought you had lost me there for a second, but by the end it totally made sense again. You do a great job at describing difficult concepts!

[haroonrasheed]

Nicely explained. I can now see how this idea can be generalised for n! + n such that infinite number of infitely large gaps would exist between primes.

[LosOnTheCoast]

I’m having trouble putting it together; How does the statement that there is a multiple of any number that is added to it prove that there are infinitely large gaps between prime numbers?

[apusapus]

There are twin primes with over 300, 000 digits, I believe. What's more important for the TPC is the nature of the agency behind these statistics, not the agency behind why some primes are so far apart.

[KyryllVasylenko-wqow]

I love your math videos ❤
They give a great opportunity to brush off dust from my brain cells and to actually think once in a while 😅

[Tabu]

Factorial makes even more sense now though.

[mattwood]

That gap is obviously not “infinitely large.”

[andraspongracz]

Infinitely large gaps? Really? What you meant was arbitrarily large finite gaps.

[Lialvas]

we need more awareness for sexy primes, everyone's talking about twin primes these days.

[rebeuhsin]

Just one infinite gap means no more primes? An infinite number of infinite gaps means, nonsense.

[Legal_Sweetie]

I should understand this no problem, but there is something about this explanation that just doesn't make sense to me.

[michi]

Aren’t and 100001!+1 prime twins? Generally speaking n!+-1?

[MohammadSami-ul-lah]

Sir i have developed a formula that allow us to figer out prime numbers.Further I have carefully observed that some mathematical tests help us to detect the total number of prime numbers that exist upto some fixed number for instance upto 50 there are 15 prime nos. and upto 100 there exist 25 primes and so on.