Voronoi congruence

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1: 24.10 Arithmetic Properties

… ►Here and elsewhere two rational numbers are congruent if the modulus divides the numerator of their difference. ►

§24.10(ii) Kummer Congruences

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§24.10(iii) Voronoi’s Congruence

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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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External Links:

ZentralBlatt

Referenced by:

§24.10(iii).

Encodings:

BibTeX

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3: 27.17 Other Applications

… ►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. …

4: 27.20 Methods of Computation: Other Number-Theoretic Functions

… ►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). …

5: Morris Newman

… ►Department of Commerce Gold Medal in 1966 for his work on algorithms for solving integral linear systems exactly by using congruence techniques. …

6: 27.15 Chinese Remainder Theorem

… ►The Chinese remainder theorem states that a system of congruences $x\equiv a_1(mod{m}_1),\mathrm{\dots },x\equiv a_k(mod{m}_k)$, 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. …

7: 27.9 Quadratic Characters

… ►If $p$ does not divide $n$, then $(n|p)$ has the value $1$ when the quadratic congruence $x^2\equiv n(modp)$ has a solution, and the value $-1$ when this congruence has no solution. …

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).

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9: 27.14 Unrestricted Partitions

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§27.14(v) Divisibility Properties

… ►After decades of nearly fruitless searching for further congruences of this type, it was believed that no others existed, until it was shown in Ono (2000) that there are infinitely many. … ►Lehmer (1947) conjectures that $\tau \left(n\right)$ is never 0 and verifies this for all by studying various congruences satisfied by $\tau \left(n\right)$, for example: …

10: Bibliography

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Congruences of $p$-adic integer order Bernoulli numbers. J. Number Theory 59 (2), pp. 374–388.

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External Links:

ISSN 0022-314X, Document, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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Congruences for the $q$-secant numbers. European J. Combin. 1 (4), pp. 283–287.

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External Links:

ISSN 0195-6698, MathReview (J. E. Cigler), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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