In this note, we show that the number of composite integers n ≤ x such that φ（n）｜n - 1 is at most O（x^1/2（loglog x）^1/2）, thus improving earlier results by Pomerance and by Shan.
In this paper,we find all positive squarefree integers d satisfying that the Pell equation X^2-d Y^2=±1 has at least two positive integer solutions(X,Y)and(X′,Y′)such that both X and X′have Zeckendorf represen...In this paper,we find all positive squarefree integers d satisfying that the Pell equation X^2-d Y^2=±1 has at least two positive integer solutions(X,Y)and(X′,Y′)such that both X and X′have Zeckendorf representations with at most two terms.展开更多
In this paper, we characterize the odd positive integers n satisfying the congruence ∑j=1^n-1 j n-1/2 We show that the set of such positive integers has an asymptotic densitywhich turns out to be slightly larger than...In this paper, we characterize the odd positive integers n satisfying the congruence ∑j=1^n-1 j n-1/2 We show that the set of such positive integers has an asymptotic densitywhich turns out to be slightly larger than 3/8.展开更多
文摘In this note, we show that the number of composite integers n ≤ x such that φ（n）｜n - 1 is at most O（x^1/2（loglog x）^1/2）, thus improving earlier results by Pomerance and by Shan.
基金supported by the project from Universidad del Valle(Grant No.71079)supported by NRF of South Africa(Grant No.CPRR160325161141)an A-Rated Scientist Award from the NRF of South Africa and by Czech Granting Agency(Grant No.17-02804S)。
文摘In this paper,we find all positive squarefree integers d satisfying that the Pell equation X^2-d Y^2=±1 has at least two positive integer solutions(X,Y)and(X′,Y′)such that both X and X′have Zeckendorf representations with at most two terms.
文摘In this paper, we characterize the odd positive integers n satisfying the congruence ∑j=1^n-1 j n-1/2 We show that the set of such positive integers has an asymptotic densitywhich turns out to be slightly larger than 3/8.
