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Power Totients with Almost Primes
-
William D. Banks
and
Florian Luca
Published/Copyright:
June 4, 2011
Abstract
A natural number n is called a k-almost prime if n has precisely k prime factors, counted with multiplicity. For any fixed k ⩾ 2, let ℱk(X) be the number of k-th powers mk ⩽ X such that φ(n) = mk for some squarefree k-almost prime n, where φ(·) is the Euler function. We show that the lower bound ℱk(X) ≫ X1/k/(log X)2k holds, where the implied constant is absolute and the lower bound is uniform over a certain range of k relative to X. In particular, our results imply that there are infinitely many pairs (p, q) of distinct primes such that (p – 1) (q – 1) is a perfect square.
Keywords.: Squares; Euler Function
Received: 2009-11-11
Accepted: 2010-02-14
Published Online: 2011-06-04
Published in Print: 2011-June
© de Gruyter 2011
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Articles in the same Issue
- Preface
- On the Intersections of Fibonacci, Pell, and Lucas Numbers
- Proving Balanced T2 and Q2 Identities Using Modular Forms
- Sum-Product Inequalities with Perturbation
- Power Totients with Almost Primes
- On the Sum of Reciprocals of Amicable Numbers
- Tromping Games: Tiling with Trominoes
- There are No Multiply-Perfect Fibonacci Numbers
- Large Zero-Free Subsets of ℤ/pℤ
Articles in the same Issue
- Preface
- On the Intersections of Fibonacci, Pell, and Lucas Numbers
- Proving Balanced T2 and Q2 Identities Using Modular Forms
- Sum-Product Inequalities with Perturbation
- Power Totients with Almost Primes
- On the Sum of Reciprocals of Amicable Numbers
- Tromping Games: Tiling with Trominoes
- There are No Multiply-Perfect Fibonacci Numbers
- Large Zero-Free Subsets of ℤ/pℤ
