Heuristics of the Cocks-Pinch method

We heuristically analyze the Cocks-Pinch method by using the Bateman-Horn conjecture. Especially, we present the first known heuristic which suggests that any efficient construction of pairing-friendly elliptic curves can efficiently generate such curves over pairing-friendly fields, naturally including the Cocks-Pinch method. Finally, some numerical evidence is given

Bounding the jj-invariant of integral points on certain modular curves

In this paper, we obtain two effective bounds for the $j$-invariant of integral points on certain modular curves which has positive genus and less than three cusps

Effective results on the Skolem Problem for linear recurrence sequences

In this paper, given a simple linear recurrence sequence of algebraic numbers, which has either a dominant characteristic root or exactly two characteristic roots of maximal modulus, we give some explicit lower bounds for the index beyond which every term of the sequence is non-zero. It turns out that this case covers almost all such sequences whose coefficients are rational numbers.Comment: 20 pages, to appear in Journal of Number Theor

On the Cycle Structure of Repeated Exponentiation Modulo a Prime Power

We obtain some results about the repeated exponentiation modulo a prime power from the viewpoint of arithmetic dynamical systems. Especially, we extend two asymptotic formulas about periodic points and tails in the case of modulo a prime to the case of modulo a prime power.Comment: Accepted by The Fibonacci Quarterly.(SCI

The Arithmetic of Carmichael Quotients

Carmichael quotients for an integer $m\ge 2$ are introduced analogous to Fermat quotients, by using Carmichael function $\lambda(m)$. Various properties of these new quotients are investigated, such as basic arithmetic properties, sequences derived from Carmichael quotients, Carmichael-Wieferich numbers, and so on. Finally, we link Carmichael quotients to perfect nonlinear functions.Comment: Proposition 4.3 is correcte

On the lattices from elliptic curves over finite fields

In this paper, we continue the recent work of Fukshansky and Maharaj on lattices from elliptic curves over finite fields. We show that there exist bases formed by minimal vectors for these lattices except only one case. We also compute their determinants, and obtain sharp bounds for the covering radius

Digraphs from Endomorphisms of Finite Cyclic Groups

We associate each endomorphism of a finite cyclic group with a digraph and study many properties of this digraph, including its adjacent matrix and automorphism group.Comment: Accepted by Journal of Combinatorial Mathematics and Combinatorial Computing. (EI

Bounding the jj-invariant of integral points on modular curves

In this paper, we give some effective bounds for the $j$-invariant of integral points on arbitrary modular curves over arbitrary number fields assuming that the number of cusps is not less than 3

On the quantitative dynamical Mordell-Lang conjecture

The dynamical Mordell-Lang conjecture concerns the structure of the intersection of an orbit in an algebraic dynamical system and an algebraic variety. In this paper, we bound the size of this intersection for various cases when it is finite.Comment: An error in Corollary 2.7 has been correcte

Bounding jj-invariant of integral points on X_{\ns}^{+}(p)

For prime $p\ge 7$, by using Baker's method we obtain two explicit bounds in terms of $p$ for the $j$-invariant of an integral point on X_{\ns}^{+}(p) which is the modular curve of level $p$ corresponding to the normalizer of a non-split Cartan subgroup of \GL_2(\Z/p\Z)