![]() ![]() ![]() We can then extend this on to quadratic sequences. However, there are a couple of ways you could look at this which combine sequences which are arguably simpler. For example, the following patterns produce a linear sequence. It got me thinking about using shapes to represent sequences and in particular using different colours to represent different sequences laid on top of each other. Given this expectation, "polynomials yield primes at different rates" reduces to "polynomials don't always have the same number of zeros mod every prime $p$," which seems (to me at least) to be not quite as surprising (since it is more finitistic, I guess).After browsing the ever excellent Don Steward’s Median site for sequences, I found this on Linear and Quadratic growths. That is to say that there may be good reason to expect that long-term "global" nature of $P(n)$ can be reliably statistically forecasted by the "local" nature of it (the counting function $N(p)$). As for intuition as to why some expressions may generate them more frequently then others (detectable in the difference in the multiplicative constants in the giant formula above), the form of the conjecture suggests that the answer may lie in local-global thinking (another thing that turns up a lot in number theory). That includes any quadratic polynomials, which are the specific subject of your question. Thus BH is a massively general statement.Įven so, I do not think any particular case of BH has been proven. It also implies the Bunyakovsky_conjecture, the one-polynomial case of BH, and is a quantitative refinement of Shinzel's hypothesis, itself an extension of Bunyakovsky to multiple polynomials. Note that a very particular case of Bateman-Horn is the quantitative version of the twin prime conjecture, the $1$st Littlewood-Hardy conjecture. "Heuristic" here - ubiquitous in analytic number theory - effectively means techniques based on experience, empirical data, intuition, educated guessing, and informal statistical reasoning. A heuristic asymptotic formula concerning the distribution of prime numbers.Then $$P(n)\sim \frac_p$ (the finite field of integers mod $p$). Let $P(n)$ count $k\le n$ for which $f_i(k)$ are prime for all $i$. Let $f_i(x)$ be a finite family of irreducible polynomials (say with positive leading coefficients). If increase the degree and number of variables, though, we do know of such polynomials that take infinitely many prime values, in fact whose positive values are only prime numbers! See this section of Wikipedia for more information.īeyond linear polynomials (the subject of the quantitative version of Dirichlet's theorems on primes in arithmetic progressions, which generalizes in a very different direction in algebraic number theory on Cheboratev density), capturing the asymptotic frequency with which systems of polynomials simultaneously yield primes is a very hard and at the moment speculative business.īateman-Horn conjecture. Whether or not we know of any quadratics that provably obtain an infinite number of prime values at integer arguments, I'm not sure. ![]() $x^2+ax+p$ for various integers $a$ and primes $p$ at $x=0$. There are also infinitely many quadratic polynomials that obtain at least one prime value, e.g. There are infinitely many linear polynomials $f(n)=an+b$ such that $f(n)$ yields prime values at infinitely many integer arguments - I assume this is what "producing primes" means.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |