If n > 3, there is always at least one prime p such that between 
.
.
An extension of this result is that if n > k, then there is a number containing a prime divisor 
in the sequence n, 
,
.
then corresponds to Bertrand's postulate.) This was first proved by Sylvester, independently by Schur, and a simple proof was given by Erdos (Erdos 1934; Hoffman 1998, p. 37)
The numbers of primes between n and 
for n = 1, 2, ... are 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 4, ... (Sloane's A077463), while the numbers of primes between n and 
are 0, 1, 1, 2, 1, 2, 2, 2, 3, 4, 3, 4, 3, 3, ... (Sloane's A060715).
A related problem is to find the least value of 
so that there exists at least one prime between n and 
for sufficiently large n (Berndt 1994). The smallest known value is 
(Lou and Yao 1992).
Choquet Theory, de Polignac's Conjecture, Prime Number
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Erdos, P. "A Theorem of Sylvester and Schur." J. London Math. Soc. 9, 282-288, 1934.
Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, 1998.
Lou, S. and Yau, Q. "A Chebyshev's Type of Prime Number Theorem in a Short Interval (II)." Hardy-Ramanujan J. 15, 1-33, 1992.
Nagell, T. Introduction to Number Theory. New York: Wiley, p. 70, 1951.
Séroul, R. Programming for Mathematicians. Berlin: Springer-Verlag, pp. 7-8, 2000.
Sloane, N. J. A. Sequences A060715 and A077463 in "The On-Line Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/.