Correction to the paper: On the sum-of-Divisors Function of the numbers of Fermat and Ferentinou-Nicolacopoulou
On the Sum-of-Divisors Function of the Numbers of Fermat and Ferentinou-Nicolacopoulou
On the system of Diophantine equations and
Perfect powers in q-binomial coefficients
Arithmetic properties of members of a binary recurrent sequence
The diophantine equation
.121221222... is not quadratic.
In this note, we show that if b > 1 is an integer, f(X) ∈ Q[X] is an integer valued quadratic polynomial and K > 0 is any constant, then the b-adic number ∑ (a / b), where a ∈ Z and 1 ≤ |a| ≤ K for all n ≥ 0, is neither rational nor quadratic.
Arithmetic properties of positive integers with fixed digit sum.
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 integer s. For example, we prove that such integers can have many prime factors, that they are not very smooth, and that most such integers have a large prime factor dividing the value of their Euler φ function.
Euler indicators of binary recurrence sequences.
Fibonacci Numbers with the Lehmer Property
We show that if m > 1 is a Fibonacci number such that ϕ(m) | m-1, where ϕ is the Euler function, then m is prime
Erratum to the Paper "Fibonacci Numbers with the Lehmer Property" (Bull. Polish Acad. Sci. Math. 55 (2007), 7-15)
On a problem of Bednarek
We answer a question of Bednarek proposed at the 9th Polish, Slovak and Czech conference in Number Theory.
On the Euler function of repdigits
For a positive integer we write for the Euler function of . In this note, we show that if is a fixed positive integer, then the equation has only finitely many positive integer solutions .
On the sum of divisors of the Mersenne numbers
On the equation
In this paper we investigate the solutions of the equation in the title, where is the Euler function. We first show that it suffices to find the solutions of the above equation when and and are coprime positive integers. For this last equation, we show that aside from a few small solutions, all the others are in a one-to-one correspondence with the Fermat primes.
On the least common multiple of Lucas subsequences
We compare the growth of the least common multiple of the numbers and , where is a Lucas sequence and is some sequence of positive integers.
On the largest prime factor of the partition function of n
Multiplicative relations on binary recurrences
Given a binary recurrence , we consider the Diophantine equation with nonnegative integer unknowns , where for 1 ≤ i < j ≤ L, , and K is a fixed parameter. We show that the above equation has only finitely many solutions and the largest one can be explicitly bounded. We demonstrate the strength of our method by completely solving a particular Diophantine equation of the above form.
How many primes can divide the values of a polynomial?
On the range of Carmichael's universal-exponent function
Let λ denote Carmichael’s function, so λ(n) is the universal exponent for the multiplicative group modulo n. It is closely related to Euler’s φ-function, but we show here that the image of λ is much denser than the image of φ. In particular the number of λ-values to x exceeds for all large x, while for φ it is equal to , an old result of Erdős. We also improve on an earlier result of the first author and Friedlander giving an upper bound for the distribution of λ-values.
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