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Two theorems on the addition of residue classes

✍ Scribed by David R. Guichard

Publisher: Elsevier Science
Year: 1990
Tongue: English
Weight: 445 KB
Volume: 81
Category: Article
ISSN: 0012-365X
DOI: 10.1016/0012-365x(90)90174-g

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✦ Synopsis

It is well known that if a,, . , a, are residues module n and m an then some sum ai, + . . .+q,, iI<...<&, is 0 (mod n). In recent related work, Sydney Bulman-Fleming and Edward T.H. Wang have studied what they call n-divisible subsequences of a finite sequence u, and made a number of conjectures. We confirm two of those conjectures in a more general form. Let f(a,, . , a,;j) be the number of sums formed from the aj which are congruent to j (mod n). We prove two main theorems: 1. Iff(a,, . . , a,,,; 0) c: 2"-' then f(q, . . . , a,; 0) s 3 . 2m-3; 2. Let m 3 n. There exist a,, . . , a,,, for which f (a,, . . , a,,,; j) is odd if and only if n is not a power of 2.

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