convexity conjecture

Conjecture (Hardy & Littlewood). Given integers xy>1, it is never the case that π(x+y)>(π(x)+π(y)), where π(x) is the prime counting function.

For example: There are 269 primes below 1729. There are 304840 primes below 4330747. If we add up these values of the prime counting function, we get 305109. This is more than π(4330747+1729)=304949.

Crandall and Pomerance believe this conjecture to be false but also that any counterexample is way too large to be discovered today. If we limit ourselves to 100 for both variables, n=π(x+y)-(π(x)+π(y)) tends to fall in the range -8<n<1.


  • 1 R. Crandall & C. Pomerance, Prime NumbersMathworldPlanetmath: A Computational Perspective, Springer, NY, 2001: 1.2.4
Title convexity conjecture
Canonical name ConvexityConjecture
Date of creation 2013-03-22 16:45:51
Last modified on 2013-03-22 16:45:51
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 7
Author PrimeFan (13766)
Entry type Conjecture
Classification msc 11A41
Synonym Hardy-Littlewood convexity conjecture