Simultaneous inequalities among values of the Euler phi-function

Details Simultaneous inequalities among values of the Euler phi-function Simultaneous inequalities among values of the Euler phi-function

2006-03-02

arxiv.org/abs/math/0603053v1

Mathematics

Number Theory

8 pages

Scientific paper

This paper concerns the values of the Euler phi-function evaluated simultaneously on k arithmetic progressions a_1 n + b_1, a_2 n + b_2, ..., a_k n + b_k. Assuming the necessary condition that no two of the polynomials a_i x + b_i are constant multiples of each other, we show that there are infinitely many integers n for which phi(a_1 n + b_1) > phi(a_2 n + b_2) > ... > phi(a_k n + b_k). In particular, there exist infinitely many strings of k consecutive integers whose phi-values are arranged from largest to smallest in any prescribed manner. Also, under the necessary condition ad \ne bc, any inequality of the form phi(an+b) < phi(cn+d) infinitely often has k consecutive solutions. In fact, we prove that the sets of solutions to these inequalities have positive lower density.

Affiliated with

Also associated with

No associations

Rating

Simultaneous inequalities among values of the Euler phi-function does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Simultaneous inequalities among values of the Euler phi-function, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Simultaneous inequalities among values of the Euler phi-function will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-369443