[math-fun] Primogeniture question
Let f(X) be a nonconstant polynomial in Z[X] not of the form f(X) = g(X) h(X) where g(X), h(X) are in Z[X] - {1,-1}. Does there necessarily exist an integer N such that f(N) is a (positive or negative) prime number ? If so, is there known to be a minimum number m(d) of such N, where d = deg(f) ? (What if f(X) is further assumed to be monic?) --Dan
"Dan" == Dan Asimov <dasimov@earthlink.net> writes:
Dan> Let f(X) be a nonconstant polynomial in Z[X] not of the form Dan> f(X) = g(X) h(X) Dan> where g(X), h(X) are in Z[X] - {1,-1}. Dan> Does there necessarily exist an integer N such that f(N) is a Dan> (positive or negative) prime number ? Dan> If so, is there known to be a minimum number m(d) of such N, Dan> where d = deg(f) ? Dan> (What if f(X) is further assumed to be monic?) As mentioned previously, this isn't known for even a single non-linear polynomial. However, there's a very recent paper by Stephan Baier and Liangyi Zhao (http://arxiv.org/PS_cache/math/pdf/0605/0605563v5.pdf) called "Primes in Quadratic Progressions on Average" which shows, in a strong sense, that it holds for "most" polynomials of the form x^2+k for k in a certain range. What it actually shows is the following: Hardy and Littlewood conjectured that sum_{n <= x} \Lambda(n^2 + k) = S(k) x + o(x), for a specific constant S(k) (which is given in the paper). Here \Lambda(n) = \log p if n = p^r, a prime power, and 0 otherwise. What the authors show is that the average of the absolute values of the remainders is "what it should be" if the conjecture holds. Victor
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Dan Asimov -
victor＠idaccr.org