Quadratic Residues and Certain Related Topics

Primitive roots, indices, quadratic residues, the Legendre symbol and the Jacobi symbol are standard topics of elementary number theory, the arithmetic in the oft-repeated quotation of Gauss which says, “Mathematics is the queen of the sciences and arithmetic the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations, she is entitled to the first rank.” The aim of this thesis is to investigate these topics each of which is concerned with the theory of congruencies and to extend the theory whenever possible. The study of primitive roots which are defined in terms of the congruence leads to an investigation of the decimal expansion of a rational number. Tables of indices are constructed and methods of construction examined. Quadratic residues and primitive roots are related and quadratic residues is used in finding primitive roots. Legendre defined a symbol which is very useful in discussions of quadratic residues and nonresidues. Jacobi later extended this definition in what is now called the Jacobi symbol. The thesis closes with a more general definition which we call the J symbol. For the benefits of the reader and for completeness we now list several definitions and notations that are used in the thesis. (1) Congruence: a ≡ b mod m or a ≡ b (m), (read “a is congruent to b modulo, or mod, m”) means m / a-b where a, b and m are integers. (2) To belong to: If e is the least positive exponent for which a^e≡ 1 (m), then a is said to belong to the exponent e modulo m. this will sometimes be abbreviated a → e (m)