Hardy-Littlewood Conjectures

Author

Eric W. Weisstein
February 3, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PrimeNumbers/Hardy-LittlewoodConjectures.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Hardy-LittlewoodConjectures.html.

©2006 Wolfram Research, Inc. except for portions noted otherwise

Counterexamples

None known

FindInstance[PrimePi[x+y]-PrimePi[x]>PrimePi[y]&&x>0&&y>0,{x,y},Integers]

FindInstance[PrimePi[x+y]-PrimePi[x]>PrimePi[y],{x,y},Integers]

PrimePi[x+y]-PrimePi[x]>PrimePi[y]/.{x->1,y->2}

Table[Sign[PrimePi[x+y]-PrimePi[x]-PrimePi[y]],{x,2,10},{y,2,10}]

With[{n=10^2},Position[Table[Sign[PrimePi[x+y]-PrimePi[x]-PrimePi[y]],{x,2,n},{y,2,x}],1]
]//Timing

With[{n=10^3},Position[Table[Sign[PrimePi[x+y]-PrimePi[x]-PrimePi[y]],{x,2,n},{y,2,x}],1]
]//Timing

ArrayPlot[Table[Sign[PrimePi[x+y]-PrimePi[x]-PrimePi[y]],{x,100},{y,100}],ColorRules->{0->White,1->Red,-1->Green}]

Position[Table[Sign[PrimePi[x+1]-PrimePi[x]-PrimePi[1]],{x,250}],1]//Flatten

Prime[Range[PrimePi[251]]]-1

Plot

With[{n=200},
Show[GraphicsArray[ListDensityPlot[Table[#,{x,n},{y,n}],ColorFunction->Hue,Mesh->False,DisplayFunction->Identity,
FrameLabel->TraditionalForm/@{x,y,#,\[Null]}
]&/@{PrimePi[x+y]-PrimePi[x],PrimePi[x]+PrimePi[y]-PrimePi[x+y]}
],ImageSize->500]
]