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Hardy-Littlewood Conjectures


The Hardy-Littlewood conjectures are two conjectures about the distribution of primes. The first is called the k-tuple conjecture. It states that the asymptotic number of prime constellations can be computed explicitly. A particular case gives the so-called strong twin prime conjecture.

The second Hardy-Littlewood conjecture states that

 pi(x+y)-pi(x)<=pi(y)

for all integers x,y>=2, where pi(x) is the prime counting function.

PrimePiDifferences

The following table summarizes the first few values of pi(x+y)-pi(x) for integer y and x=1, 2, .... The values of this function are plotted above.

yOEISpi(x+y)-pi(x) for x=1, 2, ...
1A0100511, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, ...
2A0904052, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, ...
3A0904062, 2, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 0, ...

Although it is not obvious, Richards (1974) proved that the first and second conjectures are incompatible with each other.


See also

Bouniakowsky Conjecture, Prime Constellation, Prime Counting Function, Twin Prime Conjecture

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References

Guy, R. K. §A9 in Unsolved Problems in Number Theory, 3rd ed. New York: Springer-Verlag, 2004.Hardy, G. H. and Littlewood, J. E. "Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes." Acta Math. 44, 1-70, 1923.Richards, I. "On the Incompatibility of Two Conjectures Concerning Primes; A Discussion of the Use of Computers in Attacking a Theoretical Problem." Bull. Amer. Math. Soc. 80, 419-438, 1974.Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 61-62 and 68-69, 1994.Sloane, N. J. A. Sequences A010051, A090405, and A090406 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Hardy-Littlewood Conjectures

Cite this as:

Weisstein, Eric W. "Hardy-Littlewood Conjectures." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Hardy-LittlewoodConjectures.html

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