Florian Luca

Residue Classes Having Tardy Totients (2007)

Let σ(m) be the sum of divisors of the positive integer m. Here, we show that$$\frac{1}{N+1}\sum _{0\le k\le N}\frac{\sigma...

Friedlander, John, Luca, Florian

We show, in an effective way, that there exists a sequence of congruence classes $a_k\pmod {m_k}$ such that the minimal solution $n=n_k$ of the congruence $\phi(n)\equiv a_k\pmod {m_k}$ exists and...

On the largest prime factor of the Mersenne numbers (2007)

Ford, Kevin, Luca, Florian, Shparlinski, Igor E.

Let P(k) be the largest prime factor of the positive integer k. In this paper, we prove that the series $\sum_{n\ge 1}\frac{(\log n)^a}{P(2^n-1)}$ is convergent for each constant a

On rough and smooth neighbors (2007)

Banks, William D., Luca, Florian, Shparlinski, Igor E.

Pseudoprimes in certain linear recurrences (2007)

Luca, Florian, Shparlinski, Igor

7 page(s)

Estimates for Wieferich numbers (2007)

Banks, William D, Luca, Florian, Shparlinski, Igor E

We define Wieferich numbers to be those odd integers w ≥ 3 that satisfy the congruence 2 φ(w)≡ 1(mod w²). It is clear that the distribution of Wieferich numbers is closely related to the...

On Finite fields for pairing based cryptography (2007)

Two divisors of $(n^2+1)/2$ summing up to $n+1$ (2007)

Here, we improve our previous bound on the number of finite fields over which elliptic curves of cryptographic interest with a given embedding degree and small complex multiplication discriminant may...

Ayad, Mohamed, Luca, Florian

On the square-free parts of ⌊en!⌋ (2007)

Arithmetic properties of positive integers with fixed digit sum (2006)

In this note, we show that if we write ⌊en!⌋ = s(n)u(n)², where s(n) is square-free then [equation ommitted due to formatting reasons] has at least C log logN distinct prime factors for some...

Luca , Florian

In this paper, we look at various arithmetic properties of the set of those positive integers $n$ whose sum of digits in a fixed base $b>1$ is a fixed positive integers $s$. For example, we prove...

Counting Keith numbers (2006)

Klazar, Martin, Luca, Florian

A Keith number is a positive integer N with the decimal representation a_1a_2...a_n such that n>=2 and N appears in the sequence (K_m) given by the recurrence K_1=a_1,...,K_n=a_n and...

Heron triangles with two fixed sides (2006)

Ionascu, Eugen J., Luca, Florian, Stanica, Pantelimon

In this paper, we study the function $H(a,b)$, which associates to every pair of positive integers $a$ and $b$ the number of positive integers $c$ such that the triangle of sides $a,b$ and $c$ is...

On the local behavior of the Carmichael λ-functions (2006)

Doyan, Nicolas, Luca, Florian

Arithmetic properties of the Ramanujan function (2006)

Non-Holonomicity of Sequences Defined via Elementary Functions (2006)

We study some arithmetic properties of the Ramanujan 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...

Bell, Jason P., Gerhold, Stefan, Klazar, Martin, Luca, Florian

We present a new method for proving non-holonomicity of sequences, which is based on results about the number of zeros of elementary and of analytic functions. Our approach is applicable to sequences...

Integers without divisors from a fixed arithmetic progression (2006)

Banks, William D., Friedlander, John B., Luca, Florian

Let a be an integer and q a prime number. In this paper, we find an asymptotic formula for the number of positive integers n < x with the property that no divisor d > 1 of n lies in the arithmetic...

Elliptic curves with low embedding degree (2006)

Catalan and Apéry numbers in residue classes (2006)

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...

Garaev, Moubariz Z, Luca, Florian, Shparlinski, Igor E

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 p13/2(logp)⁶ elements of each sequence already...

Common divisors of the Euler function at related arguments (2006)

Banks, William D, Luca, Florian, Shparlinski, Igor E

Let φ denote the Euler function. For a fixed integer k ≠ 0, we study positive integers n for which the largest prime factor of φ(n) also divides φ(n + k). We obtain an unconditional upper bound...

Small exponent point groups on elliptic curves (2006)

Luca, Florian, Mckee, James, Shparlinski, Igor E

Let E be an elliptic curve defined over Fq, the finite field of q elements. We show that for some constant ƞ > 0 depending only on q, there are infinitely many positive integers n such that the...

Coincidences in the values of the Euler and Carmichael functions (2006)

Banks, William D, Friedlander, John B, Luca, Florian, Pappalardi, Francesco, Shparlinski, Igor E

28 page(s)

Arithmetic properties of φ(n)/λ(n) and the structure of the multiplicative group modulo n (2006)

Banks, William D, Luca, Florian, Shparlinski, Igor E

For a positive integer n, we let φ(n) and λ(n) denote the Euler function and the Carmichael function, respectively. We define ξ(n) as the ratio φ(n)/λ(n) and study various arithmetic properties...

Elliptic curves with low embedding degree (2006)

Catalan and Apéry numbers in residue classes (2006)

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...

Garaev, Moubariz Z, Luca, Florian, Shparlinski, Igor E

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 p13/2(logp)⁶ elements of each sequence already...

Common divisors of the Euler function at related arguments (2006)

Banks, William D, Luca, Florian, Shparlinski, Igor E

Let φ denote the Euler function. For a fixed integer k ≠ 0, we study positive integers n for which the largest prime factor of φ(n) also divides φ(n + k). We obtain an unconditional upper bound...

Small exponent point groups on elliptic curves (2006)

Luca, Florian, Mckee, James, Shparlinski, Igor E

Let E be an elliptic curve defined over Fq, the finite field of q elements. We show that for some constant ƞ > 0 depending only on q, there are infinitely many positive integers n such that the...

Coincidences in the values of the Euler and Carmichael functions (2006)

Banks, William D, Friedlander, John B, Luca, Florian, Pappalardi, Francesco, Shparlinski, Igor E

28 page(s)

Arithmetic properties of φ(n)/λ(n) and the structure of the multiplicative group modulo n (2006)

Banks, William D, Luca, Florian, Shparlinski, Igor E

For a positive integer n, we let φ(n) and λ(n) denote the Euler function and the Carmichael function, respectively. We define ξ(n) as the ratio φ(n)/λ(n) and study various arithmetic properties...

Distribution of harmonic sums and Bernoulli polynomials modulo a prime (2006)

Garaev, Moubariz Z, Luca, Florian, Shparlinski, Igor E

For a fixed integer s ≥ 1, we estimate exponential sums with harmonic sums [equation omitted for formatting reasons] individually and on average, where Hs (n) is computed modulo a prime p. These...

Arithmetic properties of the Ramanujan function (2006)

On the lower bound of the linear complexity over Fp of Sidelnikov sequences (2006)

We study some arithmetic properties of the Ramanujan function τ(n), such as the largest prime divisor P(τ(n)) and the number of distinct prime divisors ω(τ(n)) of τ(n) for various sequences of...

Garaev, Moubariz Z, Luca, Florian, Shparlinski, Igor E, Winterhof, Arne

For a Sidelnikov sequence of period pm-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...

Uniform distribution of some ratios involving the number of prime and integer divisors (2006)

On the number of polynomial maps into Zn (2006)

We show that the fractional parts of the ratios n/ω(n), n/aω(n), n/τ(n) and n/aτ(n), where a ≥ 2 is a fixed integer and, as usual, ω(n) and τ(n) denote the number of prime divisors and the...

Uniformity of distribution modulo 1 of the geometric mean prime divisor (2006)

11 page(s)

Small exponent point groups on elliptic curves (2006)

We show that the fractional parts of n¹/w⁽ⁿ⁾, n¹/Ω⁽ⁿ⁾ and the geometric mean of the distinct prime factors of n are uniformly distributed modulo 1 as n ranges over all the positive...

Luca, Florian, McKee, James, Shparlinski, Igor E.

Palindromic powers (2006)

Hernández Hernández, Santos, Luca, Florian

In this paper, given an integer a > 1, we look at the smallest exponent n such that a^n is not a "palindrome".

On shifted products which are powers (2005)

On the maximal order of numbers in the "factorisatio numerorum" problem (2005)

In this note, we improve upon results of Bugeaud, Gyarmati, Sarkozy and Stewart concerning the size of a subset A of {1,...,N} such that the product of any two distinct elements of A plus 1 is a...

Klazar, Martin, Luca, Florian

Let m(n) be the number of ordered factorizations of n in factors larger than 1. We prove that for every eps>0 n^{rho} m(n) < exp[(log n)^{1/rho}/(loglog n)^{1+eps}] holds for all integers n>n_0,...

Uniform Distribution of Fractional Parts Related to Pseudoprimes (2005)

Banks, William D., Garaev, Moubariz Z., Luca, Florian, Shparlinski, Igor E.

We estimate exponential sums with the Fermat-like quotients $$ f_g(n) = \frac{g^{n-1} - 1}{n} \mand h_g(n)=\frac{g^{n-1}-1}{P(n)}, $$ where $g$ and $n$ are positive integers, $n$ is composite, and...

On Fibonacci numbers with few prime divisors (2005)

Bugeaud, Yann, Luca, Florian, Mignotte, Maurice, Siksek, Samir

If $n$ is a positive integer, write $F_n$ for the $n$th Fibonacci number, and $\omega(n)$ for the number of distinct prime divisors of $n$. We give a description of Fibonacci numbers satisfying...

.121221222... is not quadratic (2005)

Character sums and products of factorials modulo $p$ (2005)

In this note, we show that if is an integer, is an integer valued quadratic polynomial and is any constant, then the -adic number

Garaev, Moubariz Z., Luca, Florian

On the largest prime factor of $n! + 2^n - 1$ (2005)

Luca, Florian, Shparlinski, Igor E.

Prime divisors of some shifted products (2005)

Eric Levieil, Florian Luca, Igor E. Shparlinski

We study prime divisors of various sequences of positive integers A(n)+1, n=1,Ã¢Â€Â¦,N, such that the ratios a(n)=A(n)/A(nÃ¢ÂˆÂ’1) have some number-theoretic or combinatorial meaning....

Values of arithmetical functions equal to a sum of two squares (2005)

Banks, William D., Luca, Florian, Saidak, Filip, Shparlinski, Igor E.

Let ϕ(n) denote the Euler function. In this paper, we determine the order of growth for the number of positive integers n≤x for which ϕ(n) is the sum of two square numbers. We also obtain...

Values of arithmetical functions equal to a sum of two squares (2005)

Banks, William D., Luca, Florian, Saidak, Filip, Shparlinski¶, Igor E.

Let ϕ(n) denote the Euler function. In this paper, we determine the order of growth for the number of positive integers n≤x for which ϕ(n) is the sum of two square numbers. We also obtain...

Prime divisors of some shifted products (2005)

Eric Levieil, Florian Luca, Igor E. Shparlinski

We study prime divisors of various sequences of positive integers A(n)+1, n=1,…,N, such that the ratios a(n)=A(n)/A(n−1) have some number-theoretic or combinatorial meaning. In the case a(n)=n,...

Diophantine m-tuples for primes (2005)

Dujella, Andrej, Luca, Florian

We show that if p is a prime and if A = {a1,a2,…,am} is a set of positive integers with the property that aiaj+p is a perfect square for all 1 ≤ i < j ≤ m, then m < 3 · 2168. More generally,...

Values of arithmetical functions equal to a sum of two squares (2005)

Banks, William D., Luca, Florian, Saidak, Filip, Shparlinski, Igor E.

Let ϕ(n) denote the Euler function. In this paper, we determine the order of growth for the number of positive integers n≤x for which ϕ(n) is the sum of two square numbers. We also obtain...

Divisibility of class numbers: enumerative approach (2004)

Bilu, Yuri F., Luca, Florian

Murty proved that for all sufficiently large $X$ there exist at least ${c(\ell,\eps) X^{1/{4\ell}-\eps}}$ real quadratic fields with class number divisible by $\ell$ and discriminant not exceeding...

Noncototients and Nonaliquots (2004)

Banks, William D., Luca, Florian

Let $\phi(\cdot)$ and $\sigma(\cdot)$ denote the Euler function and the sum of divisors function, respectively. In this paper, we give a lower bound for the number of positive integers $m\le x$ for...

Character Sums and Congruences with n! (2004)

Garaev, Moubariz Z., Luca, Florian, Shparlinski, Igor E.

We estimate character sums with n!, on average, and individually. These bounds are used to derive new results about various congruences modulo a prime p and obtain new information about the spacings...

Exponential Sums and Congruences with Factorials (2004)

Garaev, Moubariz Z., Luca, Florian, Shparlinski, Igor E.

We estimate the number of solutions of certain diagonal congruences involving factorials. We use these results to bound exponential sums with products of two factorials $n!m!$ and also derive...

On the period of the continued fraction expansion of ${\sqrt {2^{2n+1}+1}}$ (2004)

Bugeaud, Yann, Luca, Florian

In this paper, we prove that the period of the continued fraction expansion of ${\sqrt {2^{n}+1}}$ tends to infinity when $n$ tends to infinity through odd positive integers.

Average order in cyclic groups (2004)

Von Zur Gathen, Joachim, Knopfmacher, Arnold, Luca, Florian, Lucht, Lutz G., Shparlinski, Igor E.

SUT Journal of Mathematics Vol. 41, No. 1 (2005), 31–41 On the non-trivial cycles in Collatz’s problem (2004)

Monatsh. Math. 146, 1–19 (2005) DOI 10.1007/s00605-005-0302-7 Values of the Euler Function in Various Sequences By (2004)

Abstract. In this note, we improve upon results of Steiner [9] and Mimuro [8] concerning the structure of non-trivial cycles in Collatz’s problem. AMS 2000 Mathematics Subject Classification....

William D. Banks, Kevin Ford, Florian Luca

Abstract. Let ’ðnÞ and ðnÞ denote the Euler and Carmichael functions, respectively. In this paper, we investigate the equation ’ðnÞ r ðnÞ s,wherer5s51are fixed positive integers. We also...

On the prime power factorization of n! (2003)

Luca, Florian, Stanica, Pantelimon

In this paper we prove two results. The first theorem uses a paper of Kim \cite{K} to show that for fixed primes $p_1,...,p_k$, and for fixed integers $m_1,...,m_k$, with $p_i\not|m_i$, the numbers...

A Remark on Prim Divisors of Lengths of Sides of Heron Triangles (2003)

Gaál, István, Járási, István, Luca, Florian

A Heron triangle is a triangle having the property that the lengths of its sides as well as its area are positive integers. Let {\small ${\cal P}$} be a fixed set of primes; let S denote the set of...

On the difference of values of the kernel function at consecutive integers (2003)

On the difference of values of the kernel function at consecutive integers (2003)

For each positive integer n, set ÃŽÂ³(n)=ÃŽÂ p|np. Given a fixed integer kÃ¢Â‰Â Ã‚Â±1, we establish that if the ABC-conjecture holds, then the equation...

The Diophantine equation P(x) = n! and a result of M. Overholt (2002)

For each positive integer n, set γ(n)=Πp|np. Given a fixed integer k≠±1, we establish that if the ABC-conjecture holds, then the equation γ(n+1)−γ(n)=k has only finitely many solutions. In...

Euler indicators of binary recurrence sequences (2002)

In this note, we show that the ABC-conjecture implies that a diophantine equation of the form P(x) = n! with P a polynomial with integer coefficients and degree d > 1 has only finitely many integer...

Euler indicators of binary recurrence sequences (2002)

In this paper, we show that if $(u_n)_n\geq 0$ and $(v_n)_n\geq 0$ are two non-degenerate binary recurrent sequences of integers such that $(v_n)_n\geq 0$ satisfies some technical assumptions, then...

Euler indicators of binary recurrence sequences (2002)

Instituto de Matematicas (2002)

Arnold Knopfmacher, Florian Luca, Autonoma Mexico, Lutz G. Lucht, Igor E. Shparlinski

For each natural number n we determine the average order #(n) of the elements in a cyclic group of order n. We show that a large fraction of the contribution to #(n) comes from the #(n) primitive...

Infinite sets of positive integers whose sums are free of powers (2002)

On the equation x2+2a⋅3b=yn (2002)

In this short note, we construct an infinite set S of positive integers such that for all n 1 and any n distinct elements x1, . . . , xn of S the sum n x i=1 i is not a perfect power.