The Combinatorial Bijection Implicit in Gauss on Article 148 of the Disquisitiones Arithmeticae

This preprint revisits Article 148 of Gauss’s Disquisitiones Arithmeticae, where reduced residue classes modulo a squarefree product of distinct odd primes are divided according to the parity of the number of local quadratic nonresidue conditions. Gauss states that the two resulting families have the same cardinality and refers briefly to “the theory of combinations.” The purpose of this note is to make the combinatorial content of that step explicit. We show that the omitted combinatorial layer coincides with a parity-reversing involution on subsets: fixing a distinguished element and toggling its membership. The note further lifts this combinatorial involution, via the Chinese Remainder Theorem, to an explicit arithmetic bijection between the two Jacobi classes modulo n. The construction is verified first in the minimal case and then in the example n = 105 discussed by Gauss. The contribution is structural and expository: it does not claim a new theorem beyond Gauss’s result, but makes explicit the bijection that is implicit in his counting argument.