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A characterization of exactly covering congruences

✍ Scribed by Aviezri S. Fraenkel

Publisher
Elsevier Science
Year
1973
Tongue
English
Weight
454 KB
Volume
4
Category
Article
ISSN
0012-365X

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✦ Synopsis

If every nonnegative integer occurs in exactly one of the integer sequences QiU+$, It = 0, 1,2, . . . . 0 < 41 (... Lam, 0 5 bi < aj, i = 1, . . . . m, then the system ain + bi is called an exactly covering system (ECS). Our main result is that ain f bi is an ECS if and only if I%! _ ape1 B,(bi/ai) = B, for every nonnegative integer M, where B, is the nth Bernoulli nirn!be. Several congruential identities can be derived from this result, two of which are given. It is further shown that the main result implies the well-known theorem that am_l = am for an ECS. The standard proof of this result uses a generating function and roots of unity. Two connections between this standard proof and the present proof are given at nihe end.

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