### A congruence for the second factor of the class number of a cyclotomic field

L. Carlitz (1968)

Acta Arithmetica

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L. Carlitz (1968)

Acta Arithmetica

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Lorenz Halbeisen, Norbert Hungerbühler (1997)

Journal de théorie des nombres de Bordeaux

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We give explicit non-recursive formulas to compute the Josephus-numbers $j(n,2,i)$ and $j(n,3,i)$ and explicit upper and lower bounds for $j(n,k,i)$ (where $k\ge 4$) which differ by $2k-2$ (for $k=4$ the bounds are even better). Furthermore we present a new fast algorithm to calculate $j(n,k,i)$ which is based upon the mentioned bounds.

Jerzy Urbanowicz (1990)

Compositio Mathematica

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H. Williams (1972)

Acta Arithmetica

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Jianya Liu, Tao Zhan (1997)

Acta Arithmetica

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For a large odd integer N and a positive integer r, define b = (b₁,b₂,b₃) and $(N,r)=b\in \mathbb{N}\xb3:1\le {b}_{j}\le r,({b}_{j},r)=1andb\u2081+b\u2082+b\u2083\equiv N\left(modr\right).$It is known that $(N,r)=r\xb2{\prod}_{p|r}p|N((p-1)(p-2)/p\xb2){\prod}_{p|r}p\nmid N((p\xb2-3p+3)/p\xb2)$. Let ε > 0 be arbitrary and $R={N}^{1/8-\epsilon}$. We prove that for all positive integers r ≤ R, with at most $O\left(Rlo{g}^{-A}N\right)$ exceptions, the Diophantine equation ⎧N = p₁+p₂+p₃, ⎨ ${p}_{j}\equiv {b}_{j}\left(modr\right),$ j = 1,2,3,$$⎩ with prime variables is solvable whenever b ∈ (N,r), where A > 0 is arbitrary.

W. Narkiewicz (1967)

Acta Arithmetica

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Paolo Codecà, Roberto Dvornicich, Umberto Zannier (1998)

Journal de théorie des nombres de Bordeaux

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We study two rather different problems, one arising from Diophantine geometry and one arising from Fourier analysis, which lead to very similar questions, namely to the study of the ranks of matrices with entries either zero or $\left(\right(xy/q\left)\right),(0\le x,y\<q)$, where $\left(\right(u\left)\right)=u-\left[u\right]-1/2$ denotes the “centered” fractional part of $x$. These ranks, in turn, are closely connected with the non-vanishing of the Dirichlet $L$-functions at $s=1$.

Joseph Muskat (1966)

Acta Arithmetica

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Joseph Muskat, A. Whiteman (1970)

Acta Arithmetica

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H Stark (1968)

Acta Arithmetica

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