Legendre's Conjecture (original) (raw)
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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 orsemiprime 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, ... (OEISA007491). The numbers of primes between 
and 
for 
, 2, ... are given by 2, 2, 2, 3, 2, 4, 3, 4, ... (OEIS A014085).
See also
Brocard's Conjecture, Landau's Problems
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References
Chen, J. R. "On the Distribution of Almost Primes in an Interval." Sci. Sinica 18, 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.Shannon, A. G. and Leyendekkers, J. V. "On Legendre's Conjecture." Notes Number Th. Disc. Math. 23, 117-125, 2017.Sloane, N. J. A. Sequences A007491/M1389 and A014085 in "The On-Line Encyclopedia of Integer Sequences."
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Cite this as:
Weisstein, Eric W. "Legendre's Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LegendresConjecture.html