We, know that there are infinitely many primes of the form $4n-1,4n+1,5n-1,\cdots, \text{etc}$. I saw these things in Apostol's Introduction to Analytic Number theory textbook. I would like to have an argument working for $n^{2}$. The first expression which came to my mind was $4n^{2}+3$, which gives a prime for $n=1,2,3,4,5$. For $n=6$ it gives $147$ which is divisible by 3. For $n=7$ it gives $199$ which is again a prime. Then for $n=11$, it gives $487$ which is again a prime. I would like to know whether there are infinitely many primes of the form $4n^{2}+3$? If yes, then a proof!

  • 4
    See Schinzel's hypothesis H: en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H. No result along these lines has ever been proved :-( – Robin Chapman Sep 13 '10 at 7:03
  • @Robin: Ever or never! – anonymous Sep 13 '10 at 7:31
  • 1
    @Anonymous: You made a typo in your first five n. The polynomial $F(n)=4n^2+3$ is obviously not prime for $n=3m$. – Tito Piezas III May 15 '13 at 16:42
up vote 14 down vote accepted

Many people "would like to have an argument working for $n^2$", but what is available at the moment (and for the last two centuries) are conjectures. For any list of integer polynomials there is a conjecture on how often all polynomials on the list are prime:

http://en.wikipedia.org/wiki/Bateman%E2%80%93Horn_conjecture

It is extremely hard to prove that any natural set of integers of density 0 contains infinitely many primes. It is known for the set of values of $x^2 + y^4$ but not for the values of any single-variable polynomial of degree higher than one.

The asymptotic formula in the Bateman-Horn conjecture isn't necessarily the most general expression of what people in the field believe to be true (and it is probably a lot older than Bateman and Horn's article that formally codified it), but it does subsume many earlier conjectures on primes of the form $n^2+1$, prime twins and k-tuplets, Schinzel's Hypothesis and Buniakowsy's conjecture. You can calculate from the formula the predicted frequency of $n$ such that $4n^2 + 3$ is prime.

  • @T: I don't understand you. What do you mean by saying natural set of integers of density 0 – anonymous Sep 13 '10 at 7:26
  • 1
    The set should be defined without reference to primes or to an obvious disguise for primes. Squares-plus-one is a natural set of integers for this purpose, but "integers with EulerPhi[n] divisible by (n-1)" is not natural because primality is expressible in terms of the Phi function. – T.. Sep 13 '10 at 7:40
  • 1
    Link to $x^2 + y^4$ theorem: en.wikipedia.org/wiki/Friedlander%E2%80%93Iwaniec_theorem – sdcvvc Oct 10 '11 at 15:44

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.