Is there a maximum gap between primes?

This gap has merit M = 13.1829. The largest known prime gap with identified proven primes as gap ends has length 1113106 and merit 25.90, with 18662-digit primes found by P. Cami, M. Jansen and J. K. Andersen.

How do we know that prime numbers are infinite?

A prime number is a natural number with exactly two distinct divisors: 1 and itself. Let us assume that there are finitely many primes and label them p1,…,pn. We will now construct the number P to be one more than the product of all finitely many primes: Therefore, there are infinitely many prime numbers.

Why are there an infinite number of prime numbers?

The factorial n! of a positive integer n is divisible by every integer from 2 to n, as it is the product of all of them. Hence, n! In either case, for every positive integer n, there is at least one prime bigger than n. The conclusion is that the number of primes is infinite.

Who proved that prime numbers are infinite?

Euclid
Well over 2000 years ago Euclid proved that there were infinitely many primes. Since then dozens of proofs have been devised and below we present links to several of these.

Are there any prime numbers next to each other?

The first few twin prime pairs are: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139), … OEIS: A077800.

What is the average gap between primes less than n?

That is, g (pn) is the (size of) gap between pn and pn+1. By the prime number theorem we know there are approximately n /log ( n) (natural log) primes less than n, so the “average gap” between primes less than n is log ( n ).

What is the best proof that there are infinitely many primes?

Euclid’s Proof of the Infinitude of Primes (c. 300 BC) Euclid may have been the first to give a proof that there are infinitely many primes. Even after 2000 years it stands as an excellent model of reasoning.

Are there any smaller numbers which produce the same gap?

Obviously there should be smaller numbers which produce the same gaps. For example, there is a gap of 777 composites after the prime 42842283925351–this is the least prime which produces a gap of 777 and it is far smaller than 778!+2 (which has 1914 digits). (Rather than use n !, one can also use the smaller n primorial: n #).

What is the first occurrence of a gap of at least this length?

These are the first occurrences of gaps of at least of this length. For example, there is a gap of 879 composites after the prime 277900416100927. This is the first occurrence of a gap of this length, but still is not a maximal gap since 905 composites follow the smaller prime 218209405436543 [ Nicely99 ].