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4-cycles aliquot sequence applied bipartite graph CALGARY Department complete bipartite graph complete graph COMPUTING SCIENCE condition congruence conjecture consider Corollary cycles D.H. Lehmer define the operator denote DEPARTMENT OF MATHEMATICS divisor dual integral equations edge disjoint ERDELYI exists factors field finite series formula fractional integral functions given Hall-polynomial Hanani Hankel transforms Hence implies index class inverse irreducible J.L. Selfridge J.W.T. Youngs Lemma Lowndes MATHEMATICS STATISTICS Mellin transform mod xV-l multiplier fixing Multiplier Theorem non-negative non-spreading non-trivial addition set non-trivial multiplier obtain occur partition polynomial positive integer power residues preperiod problem Proc pseudoprime Quadratic Residue reduced modulo relation Research Paper residues modulo result Ryser series in 1/r set with parameters Skolem spherical harmonics STATISTICS AND COMPUTING Steiner systems Steiner triple systems strictly divides subgraphs taken modulo Theorem 2.1 triple integral equations UNIVERSITY OF CALGARY vertex vertices xy-l