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Gauss and p-adic numbers

This paper examines Carl Friedrich Gauss's notebooks to demonstrate that his calculations involving "infinite congruences" constitute early work with p-adic numbers, including the computation of square roots and logarithms in 11-adic and 10-adic integer systems.

Original authors: Franz Lemmermeyer

Published 2026-07-30

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CC BY 4.0

Original authors: Franz Lemmermeyer

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a puzzle where the pieces don't just fit together once, but they fit together perfectly no matter how many times you zoom in. In the world of mathematics, this is the realm of number theory, a field dedicated to understanding the hidden patterns of whole numbers. Usually, when we do math, we stop at a certain point; if we need to know the square root of 2, we might say "1.414" and call it good. But what if we could keep going forever, adding more and more precise digits, not just to the right of the decimal point, but in a completely different direction? This is the world of p-adic numbers. Think of them as a special kind of number system where "closeness" is measured not by how far apart two numbers are on a number line, but by how many zeros they share at the end of their digits. If two numbers end in the same long string of zeros, they are considered very close neighbors. This might sound like a weird game, but it turns out to be a superpower for solving equations that are impossible to crack with ordinary numbers.

Now, fast forward to the early 1800s. We usually credit a German mathematician named Kurt Hensel with inventing these "infinite congruence" numbers in 1899. But a new look at the dusty, handwritten notebooks of Carl Friedrich Gauss—a genius often called the "Prince of Mathematicians"—reveals a stunning secret: Gauss was playing with these very same ideas over a century earlier. In a paper by F. Lemmermeyer, we get a peek into Gauss's private workshop, specifically his notebook from July 1800. The paper shows that Gauss wasn't just guessing; he was performing complex calculations with "infinite congruences" (which are exactly what we now call p-adic numbers) long before anyone else had a name for them. He was essentially building a toolkit for a mathematical universe that wouldn't be officially discovered until the 20th century.

The paper takes us on a tour of Gauss's specific experiments. First, Gauss was trying to find square roots and cube roots, but not the kind you find on a calculator. He was looking for roots that work perfectly within a specific "modular" world. For example, he calculated the 11-adic square root of 5, which is a number that, when squared, equals 5 in the 11-adic system (solving the equation $x^2 - 5 = 0$ ). Instead of getting a messy decimal, he found a number that, when multiplied by itself, ends in the digits of 5, no matter how many times you check. He did this by starting with a rough guess and then "lifting" it to a more precise version, digit by digit, using a method that feels like climbing a ladder where each rung gets you closer to the truth. The paper explains that Gauss used clever tricks, like the binomial theorem (a way to expand powers of numbers) and a digit-by-digit algorithm similar to the long division we learn in school, but run in reverse.

One of the most fascinating parts of the paper is how Gauss handled the number 10. Since 10 is made of 2 and 5, Gauss realized he could split the problem into two smaller, easier problems: one based on 2 and one based on 5, and then stitch them back together. This is like solving a giant jigsaw puzzle by first finishing the sky and then the ocean, and finally snapping them together. He used this to find "automorphic" numbers—numbers that, when squared, look exactly the same as the original number at the end of the line. He also calculated logarithms (a way to measure the size of numbers) in this strange 10-based system. The paper points out that while Gauss was incredibly brilliant, he wasn't perfect; the author finds a few small calculation errors in Gauss's notes, like a missing zero or a borrowed digit that wasn't accounted for. However, these mistakes don't diminish the achievement; they just show that Gauss was a human being working through complex problems in real-time.

Ultimately, this paper doesn't claim that Gauss "invented" p-adic numbers in the modern sense—he didn't write down a formal definition or a grand theory. Instead, the paper suggests that Gauss had a deep, intuitive grasp of how these infinite systems worked. He knew how to add, subtract, multiply, and even take roots of these "infinite congruences." He treated them as a practical tool to solve specific algebraic puzzles, such as finding roots of polynomials or understanding the behavior of quadratic sums. The author, Lemmermeyer, leaves us with a challenge: there are still some pages in Gauss's notebook that are hard to decipher, inviting modern readers to keep digging. The paper confirms that Gauss was a pioneer who was decades, perhaps even a century, ahead of his time, playing with the very building blocks of modern algebraic number theory while the rest of the world was still figuring out how to count.

Problem
The paper addresses the historical and mathematical gap between Carl Friedrich Gauss's unpublished calculations in his notebooks and the modern formalization of $p$ -adic numbers by Kurt Hensel in 1899. While it is well-established that Gauss possessed a form of "Hensel's Lemma" for lifting polynomial roots, the specific nature of his calculations involving "infinite congruences" (congruences modulo $p^\infty$ ) had not been fully elucidated. The author investigates Gauss's notebook entries from July 1800 to determine how he performed arithmetic with these infinite congruences, which correspond to modern $p$ -adic and $g$ -adic numbers, and to verify the accuracy and methods of his computations for square roots, cube roots, and logarithms.

Methodology
The author employs a combination of historical analysis and modern computational verification:

Historical Reconstruction: The paper analyzes specific pages from Gauss's notebooks (specifically the "1800 Juli" notebook), transcribing and interpreting his handwritten calculations.
Algorithmic Explanation: The author explains the algorithms Gauss likely used, including:
- Binomial expansions for computing roots (e.g., $\sqrt{1 \pm x}$ ).
- Digit-by-digit extraction algorithms for square roots (both standard and adapted for $p$ -adic contexts).
- Iterative approximations using the arithmetic-geometric mean inequality for cube roots.
- Power series expansions for computing $p$ -adic logarithms.
Modern Verification: The author uses the computer algebra system pari to compute modern $p$ -adic and $g$ -adic approximations (specifically modulo $241$, $11$, and $10$) to compare against Gauss's recorded digits. This includes verifying the lifting of polynomial roots, the expansion of $\sqrt{5}$ in $\mathbb{Q}_{11}$ , and the computation of 10-adic logarithms via the isomorphism $\mathbb{Z}_{10} \cong \mathbb{Z}_2 \oplus \mathbb{Z}_5$ .
Algebraic Analysis: The paper utilizes the Chinese Remainder Theorem and the theory of idempotents to explain the algebraic structure of 10-adic numbers and the existence of zero divisors and non-trivial square roots of unity.

Key Contributions

Identification of "Infinite Congruences" as $p$ -adic Numbers: The paper explicitly identifies Gauss's "infinite congruences" modulo $p^\infty$ as the precursors to Hensel's $p$ -adic numbers, demonstrating that Gauss was performing arithmetic in these rings decades before their formal definition.
Deciphering Gauss's Algorithms: The author reconstructs the specific methods Gauss used to compute:
- Square Roots: How Gauss used binomial expansions and digit-by-digit subtraction (working from right to left for $p$ -adic numbers) to find roots like $\sqrt{79}$ and $\sqrt{5}$ in $\mathbb{Q}_{11}$ .
- Cube Roots: How Gauss utilized inequalities and the arithmetic-geometric mean to approximate $\sqrt[3]{2}$ .
- Polynomial Roots: How Gauss lifted roots of a quintic polynomial from modulo 241 to modulo $241^2$ and higher powers, effectively performing Hensel lifting.
- Logarithms: How Gauss computed 10-adic logarithms using power series expansions for numbers congruent to $0 \pmod{10}$ and via factorization for coprime numbers.
Error Detection: The paper identifies specific calculation errors in Gauss's notebooks, such as a borrowing mistake in the 11-adic expansion of quadratic periods and omissions in the calculation of $\log(301)$ and $\log(799)$ .
Clarification of 10-adic Structure: The paper explains the algebraic background of Gauss's 10-adic calculations, specifically the decomposition of $\mathbb{Z}_{10}$ into $\mathbb{Z}_2 \oplus \mathbb{Z}_5$ and the role of idempotents in finding square roots of 1.

Results

Verification of Calculations: Modern computations confirm that Gauss's approximations for $\sqrt{5}$ in $\mathbb{Q}_{11}$ and the roots of the quintic polynomial modulo $241$ are correct up to the digits he recorded, despite his lack of formal notation.
Algorithmic Insight: The digit-by-digit square root algorithm used by Gauss for 10-adic numbers is shown to be a valid adaptation of standard algorithms, relying on congruences rather than magnitude.
Logarithmic Tables: The paper reconstructs a corrected table of 10-adic logarithms for integers up to 31, showing that Gauss's values generally satisfy $\log(ab) = \log(a) + \log(b)$ , though some specific entries contain arithmetic errors.
Quadratic Periods: The paper clarifies Gauss's calculation of quadratic periods related to the fifth root of unity, noting that while his method was sound, a specific arithmetic error led to incorrect values for the periods $a$ and $b$ .

Significance
The paper claims modest significance, primarily serving as an expository and detective work to "explain most of Gauss's calculations" and leave the rest as a challenge. It highlights that Gauss possessed a sophisticated, intuitive understanding of inverse limits and compatible systems of congruences, effectively performing arithmetic with $p$ -adic numbers without the formal framework later provided by Hensel. The author notes that while Gauss did not formally define these limits, his notebooks demonstrate he knew how to add, subtract, multiply, and compute roots within these systems. The paper also acknowledges that the existence of these "infinite congruences" in Gauss's work has been noted in online mathematical communities, reinforcing the idea that Gauss's toolkit was far more advanced than previously appreciated in the context of $p$ -adic analysis.

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