Irreducible values of polynomials
نویسندگان
چکیده
منابع مشابه
Irreducible Values of Polynomials: a Non-analogy
For a polynomial f(T ) ∈ Z[T ], the frequency with which the values f(n) are prime has been considered since at least the 18-th century. Euler observed, in a letter to Goldbach in 1752, that the sequence n + 1 has many prime values for 1 ≤ n ≤ 1500. Legendre assumed an arithmetic progression an+ b with (a, b) = 1 contains infinitely many primes in his work on the quadratic reciprocity law. Ther...
متن کاملA distribution problem for powerfree values of irreducible polynomials
Abstract: To better understand the distribution of gaps between kfree numbers, Erdős posed the problem of establishing an asymptotic formula for the sum of the powers of the lengths of the gaps between k-free numbers. This paper generalizes the problem of Erdős by considering moments of gaps between positive integers m for which f(m) is k-free. Here, f(x) denotes an irreducible polynomial with ...
متن کاملSHORT INTERVAL RESULTS FOR k FREE VALUES OF IRREDUCIBLE POLYNOMIALS
Let k be an integer 2. Trifonov and the author (cf. [3],[4],[15]) have recently made improvements on the gap problem of nding an h = h(x) as small as possible such that for x su ciently large, every interval of the form (x; x + h] contains a k free number. Although, one expects such h to exist with h as small as c log x for some constant c (K. McCurley and A. Zaccagnini, independent private com...
متن کاملRoots and Irreducible Polynomials
This handout, which accompanies the course on analogies between Z and F [T ], discusses some properties of polynomials in F [T ]. The results in Sections 1 and 2 work with any F , but the results in Section 3 and 4 are (somewhat) special to the field F = Fp. The main result in these notes in Theorem 3.7. The notation F [T ]h(T ), for the ring of polynomials in F [T ] considered modulo h(T ), is...
متن کاملInvariants of Defectless Irreducible Polynomials
Defectless irreducible polynomials over a valued field (F, v) have been studied by means of strict systems of polynomial extensions and complete distinguished chains. Strong connections are developed here between these two approaches and applications made to both. In the tame case where a root α of an irreducible polynomial f generates a tamely ramified extension of (F, v), simple formulas are ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Advances in Mathematics
سال: 2012
ISSN: 0001-8708
DOI: 10.1016/j.aim.2011.10.006