Florian Luca

On a conjecture about repdigits in k-generalized Fibonacci sequences

For an integer k ≥ 2, we consider the k−generalized Fibonacci sequence (F (k) n )n which starts with 0, . . . , 0, 1 (k terms) and each term afterwards is the sum of the k preceding terms. F. Luca [2] in 2000 and recently D. Marques [3] proved that 55 and 44 are the largest numbers with only one distinct digit (so called repdigits) in the sequences (F (2) n )n and (F (3) n )n, respectively. Further, Marques conjectured that there are no repdigits having at least 2 digits in a k−generalized Fibonacci sequence for any k > 3. In this talk, we report about some arithmetic properties of (F (k) n )n and confirm the conjecture raised by Marques. This is a joint work with Florian Luca.

Generalized balancing numbers

Abstract The positive integer x is a (k, l) -balancing number for y(x ≤ y — 2) if 1k + 2k + … + (x — 1)k = (x + 1)l + … + (y — 1)l for fixed positive integers k and l. In this paper, we prove some effective and ineffective finiteness statements for the balancing numbers, using certain Baker-type Diophantine results and Bilu—Tichy theorem, respectively.

On the equation x2

We find all positive integer solutions ( x , y , a , b , n ) of x 2 + 2 a ⋅ 3 b = y n with n ≥ 3 and coprime x and y .

Elliptic Curves with Low Embedding Degree

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 some heuristic arguments which suggest that there are only about z1/2+ o(1) of MNT curves with complex multiplication discriminant up to z. We also show that there are very few finite fields over which elliptic curves with small embedding degree and small complex multiplication discriminant may exist (regardless of the way they are constructed).

Fibonacci numbers at most one away from a perfect power

The famous problem of determining all perfect powers in the Fibonacci sequence and the Lucas sequence has recently been resolved by three of the present authors. We sketch the proof of this result, and we apply it to show that the only Fibonacci numbers Fn such that Fn ± 1 is a perfect power are 0, 1, 2, 3, 5 and 8. The proof of the Fibonacci Perfect Powers Theorem involves very deep mathematics, combining the modular approach used in the proof of Fermat’s Last Theorem with Baker’s Theory. By contrast, using the knowledge of the all perfect powers in the Fibonacci and Lucas sequences, the determination of the perfect powers among the numbers Fn ± 1 is quite elementary.

Exponential sums and congruences with factorials

Abstract 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 asymptotic formulas for the number of solutions of various congruences with factorials. For example, we prove that the products of two factorials n !m ! with max{n,m } < p 1/2+ ε are uniformly distributed modulo p, and that any residue class modulo p is representable in the form m !n ! + n 1! + …+ n 47! with max{m, n, n 1, … , n 47} < p 1350/1351+ ε .

Common values of the arithmetic functions ϕ and σ

We show that the equation φ(a )= σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sum-of-divisors function. This proves a fifty-year-old conjecture of Erd˝os. Moreover, we show that, for some c> 0, there are infinitely many integers n such that φ(a )= n and σ(b )= n, each having more than n c solutions. The proofs rely on the recent work of the first two authors and Konyagin on the distribution of primes p for which a given prime divides some iterate of φ at p, and on a result of Heath-Brown connecting the possible existence of Siegel zeros with the distribution of twin primes.

Irreducible radical extensions and Euler-function chains

We discuss the smallest algebraic number field which contains the nth roots of unity and which may be reached from the rational field Q by a sequence of irreducible, radical, Galois extensions. The degree D(n) of this field over Q is φ(m), where m is the smallest multiple of n divisible by each prime factor of φ(m). The prime factors of m/n are precisely the primes not dividing n but which do divide some number in the “Euler chain” φ(n),φ(φ(n)), . . . . For each fixed k, we show that D(n) > n on a set of asymptotic density 1. –For Ron Graham on his 70th birthday

Perfect fibonacci and lucas numbers

In this note, we show that the classical Fibonacci and Lucas sequence do not contain any perfect number.

Catalan and Apéry numbers in residue classes

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(log p)6 elements of each sequence already fall in all residue classes modulo every sufficiently large p, which improves the previously known result requiring pO(p) elements. We also study, using a different technique, similar questions for sequences satisfying polynomial recurrence relations like the Apery numbers. We show that such sequences form a finite additive basis modulo p for every sufficiently large prime p.