An existence proof for arbitrarily large prime gaps.

In this video we'll discuss prime gaps. In particular, we'll prove that there are arbitrarily large gaps between two consecutive prime numbers.

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This video was based on an exercise from

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[Generalth]

This feels like one of those proofs you'd have to really think about for a while before you noticed you even finished it already.

[DrWillWood]

Clarification: this proof relies on the fact that there are an infinite number of prime numbers in order to always have the larger of the two primes (separated by the prime gap) exist. Thanks to those who commented pointing this out.

[kenhaley]

You left out one important fact--there are an infinnite number of primes. For without this fact, you can actually prove the opposite. If there are only a finite number of primes, then we can find the largest gap between any two consecutivie primes, and choose any n larger than that as a counter-example to the claim.

Interestingly, the proof that there are infinitely many primes is strikingly similar to the proof given here.

[Unchained_Alice]

Great demonstration of this proof. There will likely be gaps of n with much smaller numbers each side but for that it is much harder to prove. This just proves it exists with an easy to follow proof and that is enough.

[Joffrerap]

Accurate title would be "why primes gap get arbitrarly large".

[vinzent]

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[amritawasthi]

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[koenth]

To me this feels as a simpler proof:

Let n >=3 and k be natural numbers such that 2 <= k <= n
Because k | n! we also have k | n!-k, so n!-k is composite.
As a consequence all numbers in the range [n!-n, n!-2] are composite, this is a prime gap of at least size n-1.
Since n can be chosen arbitrarily large, the prime gap can also be arbitrarily large.

[benjaminshropshire]

Nicely described. However I think to be pedantically complete it would need to prove that there will always be a next larger prime (or just mention that as a well known result).

[BrianAmedee]

That was a really cool demonstration. Such a simple explanation for a surprisingly complicated question!

[KerbalLauncher]

I think a good way to see this conceptually is with the Sieve of Eratosthenes

[annaclarafenyo]

If the primes had no large gaps, they would have density asymptotically lower-bounded from below by some real number a>0. That means for large enough N, the number of numbers less than N has to be greater than aN (the primes) + a(N/2) (the primes times 2) + a(N/3) (the primes times 3) + a(N/4) (the primes times 4) + .... , and since the sum 1/1 + 1/2 + 1/3 + 1/4 diverges, this means there will be too many numbers. This is not quite rigorous, as you need to eliminate the factors of the small numbers you use in the reciprocal-sums from you list of 'prime numbers', but this is easily remedied. This is the sieve basis of the Chebysheff method as improved by Selberg and Erdos.

[blacklistnr]

Disiked. The question in the title remains unawnswered. You treated one specific type of gaps generated by factorials. The slope of the graph and al least quantifying how big/often gaps appear were left untouched.
This is an interesting exercise with a click-baity title.

[crazedvidmaker]

In fact you only need to multiply the prime numbers less than or equal to n. Then as you add numbers less than or equal to n to the number you've composed, they cannot be prime, because they must share at least one prime factor with the number you've composed. If two numbers both share a factor, their sum has that factor. (Example: 30 comes after 29 which is prime, and the next prime isn't until 37, because 30=2*3*5, and all the numbers between 1 and 6 share at least one prime factor with 30)

[AJ-etvf]

Great interesting calming video! Love the background music. Thank you!

[xwtek]

I have a different answer. Take any series of consecutive integers. You can turn it to another series by taking LCM of every integer, turn it square free, multiply it with any integer n, and add it to every element in the series.

[jwm]

...even forty (!) years later, I'm still amazed that such a relatively huge gap of 111 exists right after the very measly prime number 370, 261. (I would end that comment with an exclamation point...except doing so would transform that measly 370261 into the way, way, way larger 370261!)

[MaxX-nl]

You can use primorial instead of factorial for smaller numbers

[jeffreyweaver]

The only Prime number that ends in 5 is the single digit 5; all other numbers that end in 5 are divisible by that single digit 5.
The only even Prime number is the single digit 2; all other even numbers other than the single digit zero that end in any of the digits 2, 4, 6, 8, or 0 are divisible by the single digit 2.
Numbers that are divisible by 3 and are composed of more than one digit can have those digits added together to make a lower number that is also divisible by 3.
All other prime numbers above 10 end in the digits 1, 3, 7, or 9, unless the number is divisible by a lower number.

[tomkerruish]

For n>3, you also have n!-n, ..., n!-2 are composite.