A note on factorization of the Fermat numbers and their factors of the form $3h2^n+1$

Michal Křížek; Jan Chleboun

Displaying similar documents to “A note on factorization of the Fermat numbers and their factors of the form $3 h 2^{n} + 1$ ”

Note on the two congruences $a x^{2} + b y^{2} + e \equiv 0$ , $a x^{2} + b y^{2} + c z^{2} + d w^{2} \equiv 0 (mod. p)$ , where $p$ is an odd prime and $a \neg \equiv 0$ , $b \neg \equiv 0$ , $c \neg \equiv 0$ , $d \neg \equiv 0 (mod. p)$

Haridas Bagchi (1949)

Rendiconti del Seminario Matematico della Università di Padova

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A note on the diophantine equation $x^{2} + b^{Y} = c^{z}$

Maohua Le (2006)

Czechoslovak Mathematical Journal

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Let $a$ , $b$ , $c$ , $r$ be positive integers such that $a^{2} + b^{2} = c^{r}$ , $min (a, b, c, r) > 1$ , $gcd (a, b) = 1, a$ is even and $r$ is odd. In this paper we prove that if $b \equiv 3 (m o d 4)$ and either $b$ or $c$ is an odd prime power, then the equation $x^{2} + b^{y} = c^{z}$ has only the positive integer solution $(x, y, z) = (a, 2, r)$ with $min (y, z) > 1$ .

A note on the congruence $(\binom{n p^{k}}{m p^{k}}) \equiv (\binom{n}{m}) (mod p^{r})$

Romeo Meštrović (2012)

Czechoslovak Mathematical Journal

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In the paper we discuss the following type congruences: $(\binom{n p^{k}}{m p^{k}}) \equiv (\binom{m}{n}) (mod p^{r}),$ where $p$ is a prime, $n$ , $m$ , $k$ and $r$ are various positive integers with $n \geq m \geq 1$ , $k \geq 1$ and $r \geq 1$ . Given positive integers $k$ and $r$ , denote by $W (k, r)$ the set of all primes $p$ such that the above congruence holds for every pair of integers $n \geq m \geq 1$ . Using Ljunggren’s and Jacobsthal’s type congruences, we establish several characterizations of sets $W (k, r)$ and inclusion relations between them for various values $k$ and $r$ . In particular, we prove that $W (k + i, r) = W (k - 1, r)$ for all $k \geq 2$ , $i \geq 0$ and...

On sets which contain a qth power residue for almost all prime modules

Mariusz Ska/lba (2005)

Colloquium Mathematicae

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A classical theorem of M. Fried [2] asserts that if non-zero integers $β ₁, . . ., β_{l}$ have the property that for each prime number p there exists a quadratic residue $β_{j}$ mod p then a certain product of an odd number of them is a square. We provide generalizations for power residues of degree n in two cases: 1) n is a prime, 2) n is a power of an odd prime. The proofs involve some combinatorial properties of finite Abelian groups and arithmetic results of [3].

Linear recurrence sequences without zeros

Artūras Dubickas, Aivaras Novikas (2014)

Czechoslovak Mathematical Journal

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Let $a_{d - 1}, \dots, a_{0} \in ℤ$ , where $d \in ℕ$ and $a_{0} \neq 0$ , and let $X = {(x_{n})}_{n = 1}^{\infty}$ be a sequence of integers given by the linear recurrence $x_{n + d} = a_{d - 1} x_{n + d - 1} + \dots + a_{0} x_{n}$ for $n = 1, 2, 3, \dots$ . We show that there are a prime number $p$ and $d$ integers $x_{1}, \dots, x_{d}$ such that no element of the sequence $X = {(x_{n})}_{n = 1}^{\infty}$ defined by the above linear recurrence is divisible by $p$ . Furthermore, for any nonnegative integer $s$ there is a prime number $p \geq 3$ and $d$ integers $x_{1}, \dots, x_{d}$ such that every element of the sequence $X = {(x_{n})}_{n = 1}^{\infty}$ defined as above modulo $p$ belongs to the set ${s + 1, s + 2, \dots, p - s - 1}$ .

Products of factorials modulo p

Florian Luca, Pantelimon Stănică (2003)

Colloquium Mathematicae

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We show that if p ≠ 5 is a prime, then the numbers $1 / p (\binom{p}{m ₁, . . ., m_{t}}) | t \geq 1, m_{i} \geq 0 f o r i = 1, . . ., t a n d \sum_{i = 1}^{t} m_{i} = p$ cover all the nonzero residue classes modulo p.

The largest prime factor of X³ + 2

A. J. Irving (2015)

Acta Arithmetica

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Improving on a theorem of Heath-Brown, we show that if X is sufficiently large then a positive proportion of the values n³ + 2: n ∈ (X,2X] have a prime factor larger than $X^{1 + 10^{- 52}}$ .

A property of the $ϕ$ and $σ_{j}$ functions

Robert E. Dressler (1975)

Compositio Mathematica

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Multiplicative functions and $k$ -automatic sequences

Soroosh Yazdani (2001)

Journal de théorie des nombres de Bordeaux

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A sequence is called $k$ -automatic if the $n$ ’th term in the sequence can be generated by a finite state machine, reading $n$ in base $k$ as input. We show that for many multiplicative functions, the sequence ${(f (n) mod v)}_{n \geq 1}$ is not $k$ -automatic. Among these multiplicative functions are $γ_{m} (n), σ_{m} (n), μ (n)$ et $φ (n)$ .

Displaying similar documents to “A note on factorization of the Fermat numbers and their factors of the form 3 h 2 n + 1 ”

Displaying similar documents to “A note on factorization of the Fermat numbers and their factors of the form $3 h 2^{n} + 1$ ”