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!
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5See Schinzel's hypothesis H: http://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H. No result along these lines has ever been proved :-( – Robin Chapman Sep 13 '10 at 07:03
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@Robin: Ever or never! – Sep 13 '10 at 07:31
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3@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
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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..
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@T: I don't understand you. What do you mean by saying natural set of integers of density 0 – Sep 13 '10 at 07:26
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1The 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 07:40
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1Link to $x^2 + y^4$ theorem: http://en.wikipedia.org/wiki/Friedlander%E2%80%93Iwaniec_theorem – sdcvvc Oct 10 '11 at 15:44