A famous conjecture (due I think to Hardy and Littlewood) states that if $P(x)$ denotes the number of primes of the form $n^2+1$ less than or equal to $x$, then $$P(x)\sim \frac{C\sqrt x}{\log x}$$ for some constant $C$ (their Conjecture F gives an expression for $C$, which I think is predicted to be somewhere in the area of $2$, although I'm not confident of that.

However, having calculated the first $1.35$ million such primes, it is not at all obvious that this is accurate. My question is the following: is there substantial numerical evidence which supports the conjecture? If so, is it simply not apparent until much higher values, or are my calculations wrong?

  • $\begingroup$ Slightly related: math.stackexchange.com/questions/879729 $\endgroup$ – punctured dusk Apr 1 '15 at 13:52
  • $\begingroup$ See numerical data for the Bateman-Horn conjecture. $\endgroup$ – Dietrich Burde Apr 1 '15 at 13:57
  • $\begingroup$ According to the prime number theorem, the number of primes of the form $x+1~($and, of course, all primes can be written in this trivial form$)$ is $\dfrac x{\ln x}$ . Replacing x with $x^2$ we have $\dfrac{\sqrt x}{\ln\sqrt x}~=~\dfrac{2~\sqrt x}{\ln x}$ . Similarly for other primes of the form $p=x^n+a$, assuming such an expression is irreducible and a is an integer. $\endgroup$ – Lucian Apr 1 '15 at 13:59
  • $\begingroup$ @Lucian Unfortunately, I don't think that can work, since, if given a prime $x+1$, $x^2+1$ is not necessarily prime. $\endgroup$ – Laertes Apr 1 '15 at 17:52
  • $\begingroup$ @Laertes: No one said that it is, but this is the basic idea. $\endgroup$ – Lucian Apr 1 '15 at 17:58

Up to $10^{24}$ there are $25814570672$ primes of the form $n^2+1,$ and the formula with $C=1.3728134628182\ldots$ predicts $24841887983$ (3.9% error). You can get better by replacing $C\sqrt x/\log x$ with $C\operatorname{li}(\sqrt{x})/2$ which gives $25814350227$ (0.00085% error). Note that $C\sqrt x/\log x \sim C\operatorname{li}(\sqrt x)/2.$

You mentioned the first 1.35 million primes of the form $n^2+1.$ The 1.35 millionth prime of that form is $1016179851635777$, and the two functions predict 1.266450 million and 1.350213 million, respectively -- not bad at all.

So yes, there is substantial numerical evidence in favor of this conjecture.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.