Residues : The gateway to higher arithmetic I
Residues to a given modulus have been introduced to mathematics by Carl Friedrich Gauss with the definition of congruence in the `Disquisitiones Arithmeticae'. Their extraordinary properties provide the basis for a change of paradigm in arithmetic. By restricting residues to remainders left over by divison Peter Gustav Lejeune Dirichlet - Gauss's successor in Göttingen - eliminated in his `Lectures on number theory' the fertile concept of residues and attributed with the number-theoretic approach to residues for more than one and a half centuries to obscure Gauss's paradigm shift in mathematics from elementary to higher arithmetic.
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