Prime gap

A prime gap is the difference between two consecutive primes. The smallest prime gap. All other prime gaps are straight, since 2 is the only even prime number and thus the difference is formed of two odd numbers. For example:, or.

Note: Some authors denote by prime gap deviation from this, the number of composite numbers between two primes, that is one less than according to the definition used here.

Occurrence of prime gaps

• Since a gap of length 1 can only occur between a straight and an odd prime number, it is obvious that they are only time. (2 is the only even prime number ).
• Whether there are infinitely many twin primes, ie, gaps of length 2 are, is one of the great unsolved problems of mathematics.
• Apart from the gap between the 2 and 3, gap length of a prime number is even.
• Since there are infinitely many prime numbers, the lengths of the prime gaps form a sequence with the initial members 1,2,2,4,2,4,2, ... ( sequence A001223 in OEIS ).

Construction of arbitrarily large prime gaps

For any natural number it is very easy to prove the existence of a prime gap of length at least. Indeed, let a natural number which is relatively prime to any of the figures. Then, the numbers are not relatively prime, and thus not prime. The largest prime number before this episode is thus at most equal to, the smallest afterwards, however, at least, so that the length of this gap is at least a prime number.

One case has different ways to make a required with the property. Evidence Technically easiest to Choose the Faculty, ie, in which case then the even considered each are divisible by. Just as well you can choose the least common multiple of the numbers from 2 to.

The smallest possible candidates are found by the Primfakultät. It is the smallest prime number greater than, the following applies, ie one even has automatically found a gap of length.

Although as small as possible in the latter case, however, there is no guarantee that the gaps are each found the first gap of the required length. In this respect, all afford these procedures, although equivalent evidence that any large gaps exist, but large gaps are of limited value in a search for the first occurrence.

Example, for n = 6

What gaps provide the procedure referred to in each case? By comparison, the first gap of length 6 occurs 23 to 29

Faculty

It is 6! = 720

So you have a prime gap of length at least 6 between the prime candidates 721 and 727 have been found. Since 721 is divisible by 7, the gap is even greater. In fact, it is framed by the prime numbers 719 and 727 and therefore has length 8

LCM (least common multiple )

It is lcm (1, ..., 6) = 60

So this time we have a gap of length at least 6 found 61-67. Both are " random" primes, that is, the length of the gap is exactly 6

Primfakultät

It is.

Again, the gap found exactly the length 6 because 31 and 37 are prime numbers.

Growth of Functions

Even the running example shows that the Faculty is by far the fastest growing among the considered functions. For the size difference is between, and even more clearly. In contrast, assigns 113-127 to a gap of length 14 so that even the so estimation is by far from being sharp.

Upper Bounds

Joseph Bertrand had the following natural boundary of a prime gap: For each applies: between and is at least a prime number. This means that a prime number gap, starting at, may not be greater than itself

It follows from the prime number theorem that the gaps are small for large against. Specifically, it can be shown that there is a constant, so that the gap after the prime is less than.

The first occurrence of a gap of a certain size

A formula that n indicates reasonably accurate for some n the first occurrence of a prime gap of length is not known.

2012 published a Japanese mathematician work, and did manifest to have found a solution for the legendary abc- conjecture. If so, the knowledge of prime numbers and prime gaps would expand enormously.