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... a more precise form of a result of C. L. Stewart ['On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers', Proc. London Math. Soc. 35(3) (1977), 425-447]. ...
2. Arithmetic properties of Apery numbers
(Uncategorized/Publication)
... an absolute constant. The method applies to more general sequences satisfying both a linear recurrence of order 2 with polynomial coefficients and certain Lucas-type congruences. ...
3. On quadratic fields generated by polynomials
(Uncategorized/Publication)
Let center dot(center dot) is an element of Z[center dot] be a polynomial of degree center dot >= 2 without multiple roots. Under the assumption of the center dot center dot center dot-conjecture, an asymptotic ...
4. On the values of the divisor function
(Uncategorized/Publication)
For a positive integer n we let tau(n) denote the number of its positive divisors. In this paper, we obtain lower and upper bounds for the average value of the ratio tau(n + 1)/tau(n) as n ranges through ...
We obtain asymptotic formulas for all the moments of certain arithmetic functions with linear recurrence sequences. We also apply our results to obtain asymptotic formulas for some mean values related ...
We show that, for most of the elliptic curves E over a prime finite field F-p of p elements, the discriminant D(E) of the quadratic number field containing the endomorphism ring of E over F-p is sufficiently ...
7. Estimates for Wieferich numbers
(Uncategorized/Publication)
We define Wieferich numbers to be those odd integers w >= 3 that satisfy the congruence 2(phi(w)) equivalent to 1 (mod w(2)). It is clear that the distribution of Wieferich numbers is closely related to ...
We estimate the number of solutions of certain congruences with Catalan numbers and middle binomial coefficients modulo a prime. We use these results to bound double exponential sums with products of two ...
Here, we improve our previous bound on the number of finite fields over which elliptical curves of cryptographic interest with a given embedding degree and small complex multiplication discrimination may ...
10. On rough and smooth neighbors
(Uncategorized/Publication)
We study the behavior of the arithmetic functions defined by F(n) = P+(n)/P-(n+1) and G(n) = P+(n + 1)/P-(n) (n >= 1), where P+(k) and P-(k) denote the largest and the smallest prime factors, respectively, ...
11. On the square-free parts of [en!]
(Uncategorized/Publication)
In this note, we show that if we write [en!] = s(n)u(n)(2), where s(n) is square-free then S(N) = Pi s(n) n 0 and sufficiently large N. A similar result is obtained for the total number of distinct primes ...
12. On the sums of complementary divisors
(Uncategorized/Publication)
In this paper, we study various arithmetic properties of d + n/d, where d runs through all the tau(n) positive divisors of n. For example, denoting by pi(n) the number of prime values among these sums, ...
For a positive integer n, we let phi(n) and lambda(n) denote the Euler function and the Carmichael function, respectively. We define xi(n) as the ratio phi(n)/lambda(n) and study various arithmetic properties ...
We study some arithmetic properties of the Ramamijan function tau(n), Such as the largest prime divisor P(tau(n)) and the number of distinct prime divisors omega(tau(n)) of tau(n) for various sequences ...
15. Catalan and Apery numbers in residue classes
(Uncategorized/Publication)
We estimate character sums with Catalan numbers and middle binomial coefficients modulo a prime p. We use this bound to show that the first at most p(13/2)(log p)(6) elements of each sequence already fall ...
For a fixed integers >= 1, we estimate exponential sums with harmonic sums H-s(n)= Sigma(n)(i=1) 1/i(s) individually and on average, where Hs(n) is computed modulo a prime p. These bounds are used to derive ...
18. Elliptic curves with low embedding degree
(Uncategorized/Publication)
Miyaji, Nakabayashi and Takano have recently suggested a construction of the so-called MNT elliptic curves with low embedding degree, which are also of importance for pairing-based cryptography. We give ...
For a Sidelnikov sequence of period p(m)-1, tight lower bounds are obtained on its linear complexity L over Fp. In particular, these bounds imply that, uniformly over all p and m, L is close to its largest ...
We show that if a > 1 is any fixed integer, then for a sufficiently large x > 1, the nth Fibonacci number F,, is a base a pseudoprime only for at most (4 + o(1)), pi(x) of positive integers n
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