I met a student that is trying to prove for fun that there are infinitely many primes of the form $n^2+1$. I tried to tell him it's a hard problem, but I lack references. Is there a paper/book covering the problem? Is this problem really hard or I remember incorrectly?
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This is an incredibly difficult problem. It is one of Landau's 4 problems which were presented at the 1912 international congress of mathematicians, all of which remains unsolved today nearly 100 years later. |
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This problem is hard in the sense that it is still unproven. I will provide a set of references, but little conclusive work (as far as I know) has been done on any of them. This is a conjecture of Hardy; he later generalized it to say: if a, b, c are relatively prime, a is positive, and $(a+b)$ and c are not both even, and $b^2 - 4ac$ is not a perfect square (I know, quite a set of conditions) - then there are infinitely many primes $an^2 + bn + c$. He does this on pg. 19 of his book. I should note that it is proved (even in the same book) that there are infinitely many primes of the form $n^2 + m^2$ and $n^2 + m^2 + 1$. (I'm pretty sure). There is another statement of this conjecture that is earlier - Are there infinitely many primes p such that p - 1 is a perfect square? This is a conjecture of Landau, and it amounts to the same thing (but without Hardy's generalization). As far as I know, the greatest work is to show that there are infinitely many numbers $n^2 + 1$ that have at most 2 prime factors, and it's pretty intense. Finally, there is a far stronger conjecture called the Horn Conjecture or the Bateman Horn Conjecture. It's a sort of generalization of many other conjectures. |
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This is a sub-problem of the Bunyakovsky conjecture. I have an interactive form of it at The Bouniakowsky Conjecture. Let $f$ be an integer-coefficient irreducible polynomial with degree higher than 2, and let $k=gcd(f(0),f(1))$. The conjecture: $f(n)/k$ always generates an infinite number of primes. Some polynomials, like $x^{12}+488669$ seem to only sparsely make prime numbers, but so far no bounds are known for any of these polynomials. |
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protected by Community♦ Jun 7 '12 at 18:44
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