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**1 - 5**of**5**### THE DIFFERENCE OF CONSECUTIVE PRIMES BY P.

"... Let pn denote the n-th prime. Backlund [1] ' proved that, for every positive e and infinitely many n, pn+i- pn> (2- e) log p.. Brauer and Zeitz [2, 10] proved that 2- e can be replaced by 4- e. Westzynthius [9] proved that for an infinity of n n+l-P n+1 pn and this was improved by Ricci [7] to> 2 lo ..."

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Let pn denote the n-th prime. Backlund [1] ' proved that, for every positive e and infinitely many n, pn+i- pn> (2- e) log p.. Brauer and Zeitz [2, 10] proved that 2- e can be replaced by 4- e. Westzynthius [9] proved that for an infinity of n n+l-P n+1 pn and this was improved by Ricci [7] to> 2 logpnlog log log p n log log log pn ' pn+i- p „> cl log pn log log log pn, where, as throughout the paper, the c's denote positive absolute constants. [4] showed that

### GAPS BETWEEN PRIME NUMBERS

, 1988

"... Let dn = Pn+i ~Pn denote the nth gap in the sequence of primes. We show that for every fixed integer A; and sufficiently large T the set of limit points of the sequence {(dn/logra, ■ • •,dn+k-i/logn)} in the cube [0, T]k has Lebesgue measure> c(k)Tk, where c(k) is a positive constant depending on ..."

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Let dn = Pn+i ~Pn denote the nth gap in the sequence of primes. We show that for every fixed integer A; and sufficiently large T the set of limit points of the sequence {(dn/logra, ■ • •,dn+k-i/logn)} in the cube [0, T]k has Lebesgue measure> c(k)Tk, where c(k) is a positive constant depending only on k. This generalizes a result of Ricci and answers a question of Erdös, who had asked to prove that the sequence {dnf log n} has a finite limit point greater than 1.