A property of the $\varphi $ and $\sigma _j$ functions

Robert E. Dressler

Displaying similar documents to “A property of the $ϕ$ and $σ_{j}$ functions”

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 $φ (n)$ .

On Sophie Germain type criteria for Fermat's Last Theorem

Andrew Granville, Barry Powell (1988)

Acta Arithmetica

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A note on factorization of the Fermat numbers and their factors of the form $3 h 2^{n} + 1$

Michal Křížek, Jan Chleboun (1994)

Mathematica Bohemica

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We show that any factorization of any composite Fermat number $F_{m} = {2^{2}}^{m} + 1$ into two nontrivial factors can be expressed in the form $F_{m} = (k 2^{n} + 1) (ℓ 2^{n} + 1)$ for some odd $k$ and $ℓ, k \geq 3, ℓ \geq 3$ , and integer $n \geq m + 2, 3 n < 2^{m}$ . We prove that the greatest common divisor of $k$ and $ℓ$ is 1, $k + ℓ \equiv 0 m o d 2^{n}, m a x (k, ℓ) \geq F_{m - 2}$ , and either $3 | k$ or $3 | ℓ$ , i.e., $3 h 2^{m + 2} + 1 | F_{m}$ for an integer $h \geq 1$ . Factorizations of $F_{m}$ into more than two factors are investigated as well. In particular, we prove that if $F_{m} = {(k 2^{n} + 1)}^{2} (ℓ 2^{j} + 1)$ then $j = n + 1, 3 | ℓ$ and $5 | ℓ$ .

Schinzel’s conjecture and divisibility of class number $h_{p}^{+}$

Stanislav Jakubec (2003)

Mathematica Slovaca

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Note on a paper of A. Rotkiewicz

T. Estermann (1963)

Acta Arithmetica

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Some fourth degree Diophantine equations in Gaussian integers.

Szabó, Sándor (2004)

Integers

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A property of the Euler $ϕ$ -function concerning the integers which are regular modulo n

Morgado, José (1974)

Portugaliae mathematica

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Displaying similar documents to “A property of the ϕ and σ j functions”

Multiplicative functions and k -automatic sequences

On Sophie Germain type criteria for Fermat's Last Theorem

A note on factorization of the Fermat numbers and their factors of the form 3 h 2 n + 1

Schinzel’s conjecture and divisibility of class number h p +

Note on a paper of A. Rotkiewicz

Some fourth degree Diophantine equations in Gaussian integers.

A property of the Euler ϕ -function concerning the integers which are regular modulo n

Displaying similar documents to “A property of the $ϕ$ and $σ_{j}$ functions”

Multiplicative functions and $k$ -automatic sequences

A note on factorization of the Fermat numbers and their factors of the form $3 h 2^{n} + 1$

Schinzel’s conjecture and divisibility of class number $h_{p}^{+}$

A property of the Euler $ϕ$ -function concerning the integers which are regular modulo n