Legendre's conjecture asserts that for every there exists a prime between and (Hardy and Wright 1979, p. 415; Ribenboim 1996,
pp. 397-398). It is one of Landau's problems.
Although it is not known if there always exists a prime between and , Chen (1975) has shown that a number which is either a prime or
semiprime does always satisfy this inequality. Moreover,
there is always a prime between and where (Iwaniec and Pintz 1984; Hardy and Wright 1979,
p. 415).
The smallest primes between and for , 2, ..., are 2, 5, 11, 17, 29, 37, 53, 67, 83, ... (OEIS
A007491). The numbers of primes between and for , 2, ... are given by 2, 2, 2, 3, 2, 4, 3, 4, ... (OEIS A014085).
Chen, J. R. "On the Distribution of Almost Primes in an Interval." Sci. Sinica18, 611-627, 1975.Hardy,
G. H. and Wright, W. M. "Unsolved Problems Concerning Primes."
§2.8 and Appendix §3 in An
Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University
Press, pp. 19 and 415-416, 1979.Iwaniec, H. and Pintz, J. "Primes
in Short Intervals." Monatsh. f. Math.98, 115-143, 1984.Ribenboim,
P. The
New Book of Prime Number Records, 3rd ed. New York: Springer-Verlag, pp. 132-134
and 206-208, 1996.Sloane, N. J. A. Sequences A007491/M1389
and A014085 in "The On-Line Encyclopedia
of Integer Sequences."
there exists a prime 
between 
and 
(Hardy and Wright 1979, p. 415; Ribenboim 1996,
pp. 397-398). It is one of Landau's problems.
between 
and 
, Chen (1975) has shown that a number 
which is either a prime or
semiprime does always satisfy this inequality. Moreover,
there is always a prime between 
and 
where 
(Iwaniec and Pintz 1984; Hardy and Wright 1979,
p. 415).
and 
for 
, 2, ..., are 2, 5, 11, 17, 29, 37, 53, 67, 83, ... (OEIS
A007491). The numbers of primes between 
and 
for 
, 2, ... are given by 2, 2, 2, 3, 2, 4, 3, 4, ... (OEIS A014085).
