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Combinatorial proof of Deddens’ theorem. (Russian) Zbl 0615.10004
Algebraic systems with one operation and one relation, Interuniv. Collect. sci. Works, Leningrad 1985, 46-53 (1985).
[For the entire collection see Zbl 0592.00016.]
Let m and n be relatively prime positive integers. Put \(G=\{um+vn\); u and v are arbitrary non-negative integers\(\}\). For any \(g\in G\) we denote by N(j,g) the number of ways of representing g as the sum of j non-zero elements of G. Put \(L(0)=1\) and \(L(g)=\sum^{\infty}_{j=1}(-1)^ j N(j,g)\) for \(g\in G\), \(g\neq 0.\)
The author gives a combinatorial proof of the following theorem of J. A. Deddens [J. Comb. Theory, Ser. A 26, 189-192 (1979; Zbl 0414.05005)]: If \(g\equiv 0\) or \(m+n\) (mod mn), then \(L(g)=1\). If \(g\equiv m\) or n (mod mn), then \(L(g)=-1\). Otherwise \(L(g)=0\).
Reviewer: B.Pondeliček

MSC:
11A07 Congruences; primitive roots; residue systems
11A25 Arithmetic functions; related numbers; inversion formulas
06F05 Ordered semigroups and monoids