Wilson’s theorem

Chandan Singh Dalawat

Displaying similar documents to “Wilson’s theorem”

Hilbert-Speiser number fields and Stickelberger ideals

Humio Ichimura (2009)

Journal de Théorie des Nombres de Bordeaux

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Let $p$ be a prime number. We say that a number field $F$ satisfies the condition $(H_{p^{n}}^{'})$ when any abelian extension $N / F$ of exponent dividing $p^{n}$ has a normal integral basis with respect to the ring of $p$ -integers. We also say that $F$ satisfies $(H_{p^{\infty}}^{'})$ when it satisfies $(H_{p^{n}}^{'})$ for all $n \geq 1$ . It is known that the rationals $ℚ$ satisfy $(H_{p^{\infty}}^{'})$ for all prime numbers $p$ . In this paper, we give a simple condition for a number field $F$ to satisfy $(H_{p^{n}}^{'})$ in terms of the ideal class group of $K = F (ζ_{p^{n}})$ and a “Stickelberger ideal” associated to the...

The number of finite groups whose element orders is given.

Moghaddamfar, A.R., Shi, W.J. (2006)

Beiträge zur Algebra und Geometrie

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Invariants and coinvariants of semilocal units modulo elliptic units

Stéphane Viguié (2012)

Journal de Théorie des Nombres de Bordeaux

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Let $p$ be a prime number, and let $k$ be an imaginary quadratic number field in which $p$ decomposes into two primes $𝔭$ and $\bar{𝔭}$ . Let $k_{\infty}$ be the unique $ℤ_{p}$ -extension of $k$ which is unramified outside of $𝔭$ , and let $K_{\infty}$ be a finite extension of $k_{\infty}$ , abelian over $k$ . Let $𝒰_{\infty} / 𝒞_{\infty}$ be the projective limit of principal semi-local units modulo elliptic units. We prove that the various modules of invariants and coinvariants of $𝒰_{\infty} / 𝒞_{\infty}$ are finite. Our approach uses distributions and the $p$ -adic $L$ -function, as defined in []. ...

On systems of composite Lehmer numbers with prime indices

J. Wójcik (1996)

Colloquium Mathematicae

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Generalized perfect numbers.

Bege, Antal, Fogarasi, Kinga (2009)

Acta Universitatis Sapientiae. Mathematica

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Representation of finite abelian group elements by subsequence sums

David J. Grynkiewicz, Luz E. Marchan, Oscar Ordaz (2009)

Journal de Théorie des Nombres de Bordeaux

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Let $G ≅ C_{n_{1}} \oplus ... \oplus C_{n_{r}}$ be a finite and nontrivial abelian group with $n_{1} | n_{2} | ... | n_{r}$ . A conjecture of Hamidoune says that if $W = w_{1} \cdot ... \cdot w_{n}$ is a sequence of integers, all but at most one relatively prime to $| G |$ , and $S$ is a sequence over $G$ with $| S | \geq | W | + | G | - 1 \geq | G | + 1$ , the maximum multiplicity of $S$ at most $| W |$ , and $σ (W) \equiv 0 mod | G |$ , then there exists a nontrivial subgroup $H$ such that every element $g \in H$ can be represented as a weighted subsequence sum of the form $g = \underset{i = 1}{\sum^{n}} w_{i} s_{i}$ , with $s_{1} \cdot ... \cdot s_{n}$ a subsequence of $S$ . We give two examples showing this does not hold in general, and characterize the...

On the number of representations in the Waring-Goldbach problem with a prime variable in an arithmetic progression

Maurizio Laporta (2012)

Journal de Théorie des Nombres de Bordeaux

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We prove a Bombieri-Vinogradov type theorem for the number of representations of an integer $N$ in the form $N = p_{1}^{g} + p_{2}^{g} + ... + p_{s}^{g}$ with $p_{1}, p_{2}, ..., p_{s}$ prime numbers such that $p_{1} \equiv l (\mod k)$ , under suitable hypothesis on $s = s (g)$ for every integer $g \geq 2$ .