# Voronoi congruence

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## 1—10 of 16 matching pages

##### 1: 24.10 Arithmetic Properties

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►Here and elsewhere two rational numbers are

*congruent*if the modulus divides the numerator of their difference. ►###### §24.10(ii) Kummer Congruences

… ►###### §24.10(iii) Voronoi’s Congruence

…##### 2: Bibliography P

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Voronoi type congruences for Bernoulli numbers.
In Voronoi’s Impact on Modern Science. Book I, P. Engel and H. Syta (Eds.),
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##### 3: 27.17 Other Applications

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►Congruences are used in constructing perpetual calendars, splicing telephone cables, scheduling round-robin tournaments, devising systematic methods for storing computer files, and generating pseudorandom numbers.
…Apostol and Zuckerman (1951) uses congruences to construct magic squares.
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##### 4: 27.20 Methods of Computation: Other Number-Theoretic Functions

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►A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function $\tau \left(n\right)$, and the values can be checked by the congruence (27.14.20).
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##### 5: Morris Newman

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►Department of Commerce Gold Medal in 1966 for his work on algorithms for solving integral linear systems exactly by using congruence techniques.
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##### 6: 27.15 Chinese Remainder Theorem

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►The Chinese remainder theorem states that a system of congruences
$x\equiv {a}_{1}\phantom{\rule{0.949em}{0ex}}(mod{m}_{1}),\mathrm{\dots},x\equiv {a}_{k}\phantom{\rule{0.949em}{0ex}}(mod{m}_{k})$, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod $m$), where $m$ is the product of the moduli.
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##### 7: 27.9 Quadratic Characters

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►If $p$ does not divide $n$, then $(n|p)$ has the value $1$ when the quadratic congruence
${x}^{2}\equiv n\phantom{\rule{0.949em}{0ex}}(modp)$ has a solution, and the value $-1$ when this congruence has no solution.
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##### 8: 24.19 Methods of Computation

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Tanner and Wagstaff (1987) derives a congruence $(modp)$ for Bernoulli numbers in terms of sums of powers. See also §24.10(iii).

##### 9: 27.14 Unrestricted Partitions

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