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I was reading Martin Gardner's Mathematical Games column on the Ulam spiral which appeared in the March 1964 issue of Scientific American. (The spiral actually featured on the cover of that issue.) Gardner makes the following statement:

The grid on the cover suggests that throughout the entire number series expressions of this form are likely to vary markedly from those "poor" in primes to those that are "rich," and that on the rich lines an unusual amount of clumping occurs.

By "this form" Gardner means the form $4x^2+bx+c$. I'm curious - and a little bit skeptical - about his last statement concerning clumping. I know that the existence of prime-rich and prime-poor polynomials is a longstanding conjecture, going back to Euler's discovery that polynomials such as $x^2-x+41$ generate unusually many primes, and that Hardy and Littlewood and also Bateman and Horn made concrete proposals as to what the density of primes in such polynomials ought to be.

My question is whether there is any evidence, either numerical or heuristic, that there should be a large amount of clumping in the primes of the form $x^2-x+41$. Famously, the first 40 values of $x$ all give primes, but if one goes to higher values of $x$ are there more long clusters of primes than one would expect if the primes were randomly distributed?

Rephrasing the question: I am aware of the conjecture that $x^2-x+41$ has more primes than other, similar lines. The question is whether there is a conjecture saying that $x^2-x+41$ has more dense clusters of primes than expected.

3 Answers 3

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You will find this paper of interest:

Fung and Williams, "Quadratic polynomials which have a high density of prime values", Mathematics of Computation, 55:191, July 1990, 345-353.

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    Thanks for the suggestion. That's a very nice paper, which I actually had run across previously. I'll have to take a closer look so see whether they address the clumping issue.2010-11-21
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It is easy to check numerically. For $x^2-x+41$ I found the following values:

x $\leq$ - number of primes - number of expected primes - ratio

$1000000 - 261082 - 39313 - 6.64$

$5000000 - 1157818 - 174318 - 6.64$

For $x^2-x+43$ :

$1000000 - 49233 - 39313 - 1.25$

$5000000 - 219098 - 174318 - 1.25$

For $x^2-x+45$ :

$1000000 - 32060 - 39313 - 0.81$ $ 5000000 - 141501 - 174318 - 0.81$

For the expected number of primes I summed up the probabilities that a number x (that is choosen randomly) is prime, $\frac{1}{ln(x)}$. Not only are there much more primes in $x^2-x+41$ than the other two polynomials, the ratio as I calculated it looks like it's converging. Withoug having read the whole paper Matthew Conroy linked, I'm pretty sure that the ratios are in fact the Hardy-Littlewood constants.

Edit: I see what you mean now. I'm trying to find a mathematical definition of "clumping", but it is all somewhat vage...

Edit2: Perhaps this is an approach: We start small with the number of expected "twinprimes", which in our case means $f(x)=x²-x+41$ is prime for consecutive integers. The probability should be about $\sum \frac{6.64^2}{\ln{f(x)}\ln{f(x+1)}}$. The $6.64$ is due to the fact that our polynomial has about that much more primes than normally expected.

$1000000 - 69152 - 68885 - 1.003870966737562$

$2000000 - 124384 - 123599 - 1.0063466049598313$

$3000000 - 175873 - 174474 - 1.008017081247298$

$4000000 - 225335 - 223075 - 1.0101310763434574$

$5,000,000 - 273080 - 270083 - 1.011093447429278$

It is pretty close to what I expected. Perhaps you need to look at bigger clumps to see a big difference.

$100,000,000 - 3723447 - 3678470 - 1.0122270933045168$

$200,000,000 - 6877502 - 6797647 - 1.0117473483885233$

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    It would make things worse for my expectations that the probabilities for$f(x)$and$f(x+1)$are independent. But it would certainly make the whole thing more interesting :)2014-06-30
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Clumps of primes from such quadratic equations should behave similarly to variations in the gaps between all primes. In general, richer functions in primes are more likely to find groups. There are other quadratics that generate streaks of consecutive outputs that are prime, such as:

2n^2 - 272431: Prime for n = 371 to 393 (23 in a row) 2n^2 + 144251: Two streaks, n = 34 to 50 (17) and n = 583 to 602 (20). n(n+1) - 1776433: Prime for n = 1424 to 1443 (20); my JVM has calculated the prime density of this one at about 8.32 (versus Euler's ~6.64). Its smallest dividing primes are 41, 59, 97, and 101. It's nowhere near the records for the richest functions, though.

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    The difficulty of generating long streaks of prime outputs is greater for larger numbers - quadratic equations with prime densities higher than 9 times average tend to have much larger discriminants and can't yield many small outputs in a row. n(n+1) - 1776433 is still the richest equation in primes that I've found to have a streak of 20 or longer.2017-02-16