Proof by Reduction

Oswald Baumgart

Chapter Chapter 3 in The Quadratic Reciprocity Law, 2015, pp 15-39 from Springer

Abstract: Abstract 1. If q denotes a prime number, then $$1,2,\ldots, \frac{q-1} {2}$$ is a complete system of incongruent positive minimal1 residues modulo q; on the other hand, if a is coprime to q, then a, 2a, …, $$\frac{q-1} {2} a$$ is a system of $$\frac{q-1} {2}$$ incongruent residues which do not necessarily form a half-system modulo q. If, in this last set, ρ 1, …, ρ λ are the positive and −σ 1, …, −σ μ the negative minimal residues modulo q, then we can observe that the ρ and σ are nonzero and pairwise distinct, hence congruent modulo q in some order to the numbers $$1,2,\ldots, \frac{q-1} {2}$$ .

Keywords: Congruent Modulo; Minimal Residue; Beautiful Lemma; System Half; Relations Hold (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/978-3-319-16283-6_3

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