what is the biggest interval between two prime numbers?

Answer 1

It can be as long as you want.
Choose whatever positive integer n you want. Consider these numbers: n! + 2, n! + 3, .. , n! + n. They are n consecutive numbers. The first one is divisible by 2 (since n! = n*..*2*1, and 2 is), the second one is divisible by 3, .., the last is divisible by n. So none of those are prime.
So, whatever n you chose, you have n-1 consecutive non-primes. If I choose n = 1000001, theres a gap of at least 1000000 numbers between any primes there.

Answer 2

stephen m is correct - however large an interval you would like, you can write down two numbers which are that far apart and which have no prime number between them. You will not know which is the last prime number before the gap, or the first prime number after it, but whatever they are, the gap is at least that big.

The largest gap between actually known primes, where it is also known that all smaller primes have smaller gaps, is a gap of 1356 starting at the 18-digit prime 401429925999153707. It was discovered earlier this year by Donald E. Knuth. Among numbers of this size, every 40th number on average is prime.

Answer 3

Nobody has ever found a definite answer to the distribution of prime numbers. However, the intervals between prime numbers does tend to grow as the prime numbers grow. But to go short, there is no definate answer to this. Essentially, the intervals will go to infinity as the prime number increases.

Answer 4

A prime gap can be as large as you want it to be; n!+1 to n!+n+1 is the classic example of a gap of length at least n.

The largest known prime gap, as bh8153 correctly points out, is 1356: 401429925999153707 to 401429925999155063.

In general, the average gap size near n is log (n), and the largest gap up to n is generally about* log(n)^2.

* This is conjectured (Legendre's conjecture), but not proved. This fits known primes very closely, but what can be proved at the moment is much weaker.

Answer 5

There's no known correct answer for this, because we don't know what the highest possible prime number is (the largest known at this time is (2^13,466,917) - 1), but the answer would be whatever the interval is between the largest prime number and the second-largest one.

Answer 6

1 is the smallest positive prime number.
There is largest prime number. It has been proven that for any prime number there is always a larger prime number. Therefore the biggest interval is infinity -1 = infinity.

Answer 7

this is impossible to know. the prime number list is infinite and the interval is bigger and bigger when you go through the list.
maybe the interval tends to be infinite too

Answer 8

infinity