## You are here

Homeconvexity conjecture

## Primary tabs

# convexity conjecture

Conjecture (Hardy & Littlewood). Given integers $x\geq y>1$, it is never the case that $\pi(x+y)>(\pi(x)+\pi(y))$, where $\pi(x)$ 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 $\pi(4330747+1729)=304949$.

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, $n=\pi(x+y)-(\pi(x)+\pi(y))$ tends to fall in the range $-8<n<1$.

# References

- 1 R. Crandall & C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001: 1.2.4

Synonym:

Hardy-Littlewood convexity conjecture

Type of Math Object:

Conjecture

Major Section:

Reference

## Mathematics Subject Classification

11A41*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections

## Comments

## Why convex?

Why is this called "convexity conjecture"? I've read the reference, too, and they don't explain it at all.

## Re: Why convex?

> Why is this called "convexity conjecture"? I've read the

> reference, too, and they don't explain it at all.

Here's a first approximation: if $\pi$ were a convex function,

then it would follow that

\[

\pi( (x+y)/2 ) \le (\pi(x) + \pi(y)) / 2.

\]

But \pi( (x+y)/2 ) doesn't make sense if $x+y$ is odd, so

let's drop the 2s:

\[

\pi(x + y) \le \pi(x) + \pi(y).

\]

I'm a bit uneasy about this, since $\pi(x/2)$ is not the same

as $\pi(x)/2$, but that may also be the reason Crandall and

Pomerance put ``convexity'' in scare quotes.

I hope an analyst will pop in to clear things up.

## Re: Why convex?

Do they say who originally came up with this conjecture.

Presumably, to find out why the term "convexity conjecture",

you need to know who came up with the term and then see how

that person justified the term. If that person didn't say

anything, then Michael Slone's guess seems as good as any.

## Re: Why convex?

OOps, you already do know who came up with it. So then the

question becomes to to figure out in where Hardy and Littlewood

first stated this conjecture, read the reference, and see what

they had to say. Of course, it may happen that they weren't

the ones who termed it "convexity conjecture", in which case

you might need to spend time in the library tracking down forward

references to their original article to figure out who exactly

came up with the term and why. At that point, it becomes an

issue of whether the question is interesting enough to bother

getting to the bottom ofthe issue.

## Re: Why convex?

Think easier, think progressive.

Hardy and Littlewood lived in simpler times, when mathematicians weren't as concerned as they are today with piling up abstractions and generalizations like floors on a house of cards.

They were able to visualize things like the Julia fractals. So what if we take PrimeFan's n and change it to z (an axis)? It then becomes very easy to visualize what kind of shape would result if this conjecture was to be proven false and we plotted at the coordinates given by the counterexample.

But don't take my word for it, put it through your favorite CAS. In Mathematica, you can play around with

Plot3D[PrimePi[x + y] - (PrimePi[x] + PrimePi[y]), {x, 1, 5000}, {y, 1, 5000}]

or

Plot3D[PrimePi[x] + PrimePi[y], {x, -2500, 2500}, {y, -2500, 2500}]

etc.

## Re: Why convex?

I don't think anyone here would take your word for it if you said 1 plus 1 is 2 (giggle). But seriously, you make a good point about the helpfulness of plotting tools. Per Cran & Pom, I'm gonna try taking these plots to x = y = 20000 if my computer can handle it.

## Re: Why convex?

> At that point, it becomes an

issue of whether the question is interesting enough to bother

getting to the bottom ofthe issue.

For me it is interesting enough, plus it gives me another reason to go down to the library (when I just have one thing to look up, it's just not a priority with everything else I have going -- now I have this plus the Lucas-Lehmer code and a couple of topics Bob Happ asked me about.