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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

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 ≥ 3 , ℓ ≥ 3 , and integer n ≥ m + 2 , 3 n < 2 m . We prove that the greatest common divisor of k and ℓ is 1, k + ℓ ≡ 0 m o d 2 n , m a x ( k , ℓ ) ≥ F m - 2 , and either 3 | k or 3 | ℓ , i.e., 3 h 2 m + 2 + 1 | F m for an integer h ≥ 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 .

A note on some discrete valuation rings of arithmetical functions

Emil Daniel Schwab, Gheorghe Silberberg (2000)

Archivum Mathematicum

The paper studies the structure of the ring A of arithmetical functions, where the multiplication is defined as the Dirichlet convolution. It is proven that A itself is not a discrete valuation ring, but a certain extension of it is constructed,this extension being a discrete valuation ring. Finally, the metric structure of the ring A is examined.

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