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1. Introduction. The recent article [1] gives explicit evaluations for exponential sums of the form $S (a, p^{α} + 1) : = \sum_{x \in_{q}} χ (a x^{p^{α} + 1})$ where χ is a non-trivial additive character of the finite field $_{q}$ , $q = p^{e}$ odd, and $a \in *_{q}$ . In my dissertation [5], in particular in [4], I considered more generally the sums S(a,N) for all factors N of $p^{α} + 1$ . The aim of the present note is to evaluate S(a,N) in a short way, following [4]. We note that our result is also valid for even q, and the technique used in our proof can also be used to evaluate certain...

A note on exponential sums.

L. Carlitz (1978)

Mathematica Scandinavica

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 $ℓ, k \geq 3, ℓ \geq 3$ , 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 $5 | ℓ$ .

Currently displaying 521 – 540 of 1970

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Displaying 521 – 540 of 1970

A note on colored Tornheim's double series.

A note on consecutive ones in a binary matrix.

A note on counting cuspidal excursions.

A Note on Cyclotomic Fields.

A note on Deaconescu's result concerning Lehmer's problem.

A note on Diophantine approximation.

A note on Dirichlet's L-functions

A Note on Eisenstein΄s Criterion

A note on elliptic functions and approximation by algebraic numbers of bounded degree

A note on Euler number and polynomials.

A note on evaluations of some exponential sums

A note on exponential sums.

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

A note on Farey fractions with denominators in arithmetic progressions

A note on Fibonacci matrices of even degree.

A note on Fibonacci-type polynomials.

A note on fields of definition

A note on formal groups and zeta functions.

A note on Format's conjecture

A Note on Formulae for the nth Prime.

Currently displaying 521 – 540 of 1970