# What is the largest gap between primes?

## What is the largest gap between primes?

Numerical results As of September 2017, the largest known prime gap with identified probable prime gap ends has length 6582144, with 216841-digit probable primes found by Martin Raab. This gap has merit M = 13.1829.

### Do prime numbers get farther apart?

An interesting aspect of prime numbers is they come farther and farther apart as more are found—except sometimes, they don’t—sometimes instead, they come in pairs: 11 and 13 for example, or 41 and 43. The twin prime conjecture states that there are infinitely many pairs, but no one has been able to prove it.

#### How far apart are primes?

Zhang, of the University of New Hampshire, showed for the first time that even though primes get increasingly rare as you go further out along the number line, you will never stop finding pairs of primes that are a bounded distance apart — within 70 million, he proved.

What is the maximum difference between two prime numbers?

There is only one distinct prime number so the maximum difference would be 0. There is no prime number in the given range so the output for the given range would be -1.

How fast do primes grow?

So if the computing power available for seeking primes doubles every k months, then the size of the largest known prime should double every 3k months. The slope 0.079 (over past 60 years) corresponds to doubling the digits every 3.8 years, or 46 months.

## Can prime numbers other than 2 and 5 ever be 3 apart?

Because the difference of the prime numbers is 2, then the two numbers subtracted from one another must be even-odd or odd-even. Because the only even prime number is 2, the two prime numbers whose difference is 3 are 5 and 2. 2 and 5 are the only two primes whose difference is 3.

### Is there any relation between prime numbers?

Prime numbers are numbers that can only be evenly divided by 1 and themselves, such as 5 and 17. If A and B are two such numbers and C is their sum, the ABC conjecture holds that the square-free part of the product A x B x C, denoted by sqp(ABC), divided by C is always greater than 0.

#### Are prime gaps bounded?

But what Yitang Zhang just proved is that there are infinitely many pairs of primes that differ by at most 70,000,000. In other words, that the gap between one prime and the next is bounded by 70,000,000 infinitely often—thus, the “bounded gaps” conjecture. On first glance, this might seem a miraculous phenomenon.

Are 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. for some natural number n; that is, the number between the two primes is a multiple of 6.

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 ).

## 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 ].

#### What is the least prime number which produces a gap of 777?

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 #).

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