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GAPS BETWEEN PRIME NUMBERS. ADOLF HILDEBRAND AND HELMUT MAIER. (Communicated by Larry J. Goldstein). ABSTRACT. Let dn = Pn+1 -Pn denote the nth gap in the ...

These are 122, 123, 124, 125. The first is divisible by 2, the second divisible by 3, the third divisible by 4, and the last divisible by 5. So here we have four consecutive non‐primes, meaning that we have a gap of at least four (4) between two consecutive primes. Three comments are in order related to this result and the proof above.

The first, smallest, and only odd prime gap is the gap of size 1 between 2, the only even prime number, and 3, the first odd prime. All other prime gaps are even. There is only one pair of consecutive gaps having length 2: the gaps g2 and g3 between the primes 3, 5, and 7. For any integer n, the factorial n! is the product of all positive integers up to and including n. Then in the sequence

Prime gap Simple observations. The first, smallest, and only odd prime gap is the gap of size 1 between 2, the only even prime... Numerical results. Usually the ratio of gn / ln ( pn) is called the merit of the gap gn . ... This gap has merit M = 13. Further results. Bertrand's postulate, proven in ...

For over a century, mathematicians have understood how the primes taper off on average: Among large numbers, the expected gap between prime ...

As numbers get larger, there are more possible divisors, so the chance of a randomly chosen number being prime decreases. Thus, the average ...

Some define the gap between these two primes to be the number of composites between them, so g = q - p - 1 (and the gap following the prime 2 has length 0). Others define it to be simply q - p (so the gap following the prime 2 has the length 1). On these pages we use the former definition. Jens Kruse Andersen's page on maximal gaps and Nicely's pages use the second.

So letting pn be the n th prime we have: pn+1 = pn + g ( pn) + 1. 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 ).

GAPS BETWEEN PRIME NUMBERS3 LEMMA 1. Let q be a good modulus. Then we have, uniformly for x > qD and (a,q) = 1, X n(2x,q,a) -Tr(x,q,a) » ——-. £>(<?) log x Here D is a constant > 1 that depends only on the constant C implicit in the definition of a good modulus. We shall apply this result with moduli q of the form p(z)=Hp. p<z

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 ...

A prime gap of length is a run of consecutive composite numbers between two successive primes. Therefore, the difference between two successive primes and bounding a prime gap of length is , where is the th prime number. Since the prime difference function (1) is always even (except for ), all primes gaps are also even.

The prime number theorem says that the number of primes ≤x ≤ x is asymptotically x/logx x / log ⁡ x as x→∞ x → ∞ . This was proved in 1896 by Hadamard and ...

Some define the gap between these two primes to be the number of composites between them, so g = q - p - 1 (and the gap following the prime 2 has length 0). Others define it to be simply q - p (so the gap following the prime 2 has the length 1). On these pages we use the former definition.

On the one hand there's all these reports of the largest prime gap between all numbers proven to be < 246. On the other hand, I read reports of ...

A prime gap of length is a run of consecutive composite numbers between two successive primes. Therefore, the difference between two successive primes and bounding a prime gap of length is , where is the th prime number. Since the prime difference function (1) is always even (except for ), all primes gaps are also even.

ture of the prime numbers, one would expect any gap between consecutive prime numbers to occur, and nevertheless in nitely often. Now let us compose one of these intervals: Let n2N Consider n! = 1 2 :::(n 1) n)8i= 1;2;:::;n: njn!)8i= 2;:::;n: n+ i- n! + i)8m2[n! + 2;n! + n];m=2P Therefore; there are gaps between prime numbers of size at least n-1. 5

The largest gap between successive prime numbers is unbounded. The larger the prime number, the larger the average gap between it and its successor or ...

A prime gap of length n is a run of n-1 consecutive composite numbers between two successive primes. Therefore, the difference between two successive primes ...