Power Totients with Almost Primes
نویسندگان
چکیده
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 m 6 X such that !.n/ D m for some squarefree k-almost prime n, where !. ! / is the Euler function. We show that the lower bound Fk.X/ " X=.log X/ 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.
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تاریخ انتشار 2011