Gauss and p-adic numbers

2025/10/13 by Franz Lemmermeyer, Lemmermeyer, Franz
Mathematics · #01A55 #FOS: Mathematics #History and Overview (math.HO) #Number Theory (math.NT) #math.HO #math.NT #msc:01A55

paper · pdf · doi:10.48550/arxiv.2510.11559

arxiv created 2026/07/29 · arxiv updated 2026/07/30

Abstract

In his notebooks, Gauss recorded various calculations with "infinite congruences". These infinite congruences are p-adic numbers; Gauss computes a square root of 5 in the 11-adic integers in order to find an 11-adic approximation to a quadratic Gauss sum, computes a nontrivial square root of 1 in 10-adic integers, and computes the 10-adic logarithms of small natural numbers.

Related

• A brief introduction to p-adic numbers
• Notes on p-adic numbers
• The last chapter of the Disquisitiones of Gauss
• Walking to infinity on gaussian lines
• Binary quadratic forms and the factorization method of Gauss