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On Some Generalisations of Gauss Sequences

This paper introduces and investigates "Euler–Gauss sequences," a generalization of Gauss sequences that incorporates distinct prime factors and admits $q$ -analogs satisfying the Cyclic Sieving Phenomenon, while also providing new divisibility criteria and counterexamples to existing conjectures.

Original authors: Sathyanarayan Narayan, N. Uday Kiran

Published 2026-08-21

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CC BY 4.0

Original authors: Sathyanarayan Narayan, N. Uday Kiran

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

Mathematicians have long been fascinated by the hidden patterns that govern whole numbers. Among the most enduring of these patterns are rules that describe how numbers behave when divided, known as congruences. One famous rule, discovered by Leonhard Euler, explains how certain powers of a number relate to the number itself when the two share no common factors. A later refinement by Carl Friedrich Gauss tightened this rule, creating a stricter set of conditions that only specific sequences of numbers could satisfy. These "Gauss sequences" are not just abstract curiosities; they appear in diverse areas of mathematics, from the study of prime numbers to the analysis of complex geometric shapes. For decades, researchers have tried to map the boundaries of these rules, asking which number patterns fit the strict Gauss definition and which ones fall just outside it.

In a recent paper, mathematicians Sathyanarayan Narayan and N. Uday Kiran from the Sri Sathya Sai Institute of Higher Learning in India have drawn a new map of this territory. They introduce a fresh category of number sequences they call "Euler–Gauss sequences." Think of these as a middle ground: they are broader than the strict Gauss sequences but still follow a specific, elegant rule inspired by Euler's original work. The researchers discovered that while every Gauss sequence fits into this new category, the reverse is not true. There are many interesting number patterns that satisfy the new, slightly looser rule but fail the older, stricter test. This distinction is crucial because it reveals a wider universe of mathematical structures that were previously overlooked or lumped together incorrectly.

The authors did not just define this new group; they filled in the gaps of existing knowledge by showing exactly where the old rules break down. For instance, they examined sequences based on the smallest and largest prime factors of a number. In the world of integers, every number greater than one is built from prime numbers, which are the indivisible building blocks of arithmetic. The smallest prime factor is the first prime that divides a number, while the greatest prime factor is the largest. The researchers showed that sequences tracking these specific factors fit perfectly into their new Euler–Gauss category. However, these same sequences do not satisfy the older, stricter Gauss rules. This finding is significant because it proves that the new category captures important mathematical behaviors that the old definitions missed. It suggests that the landscape of number patterns is more nuanced than previously thought, with a distinct layer of sequences that obey a unique set of congruence properties.

To make their findings even more powerful, the researchers extended these ideas into a realm called "q-analogs." In mathematics, a q-analog is a way of generalizing a standard rule by introducing a variable, often denoted as q, which allows the rule to behave differently depending on how it is tuned. When this variable is set to a specific value, the rule usually reverts to the standard integer version. The team defined "q-Euler–Gauss sequences" and showed that they, too, form a distinct class. They proved that while every standard q-Gauss sequence is also a q-Euler–Gauss sequence, the new class includes many more examples. They even identified a specific condition under which a q-Euler–Gauss sequence becomes a q-Gauss sequence: if the sequence never hits zero when the variable is set to one. If the sequence does hit zero, it remains in the broader category and cannot be forced into the stricter class. This clarification helps resolve confusion in the literature about how these different types of sequences relate to one another.

One of the most exciting outcomes of this work is a new connection to a phenomenon known as the Cyclic Sieving Phenomenon. This is a way of counting objects in a set that change when rotated, where the number of objects that stay the same depends on how far they are rotated. The researchers showed that their new sequences, particularly those based on the smallest and largest prime factors, generate sets of objects that follow this phenomenon in a unique way. Unlike the standard patterns seen in older sequences, these new sequences follow a different counting rule that depends on the specific prime factors of the numbers involved. This provides a fresh combinatorial interpretation for these sequences, linking abstract number theory to the concrete counting of physical or geometric arrangements.

The paper also addresses several misconceptions and fills holes in previous studies. The authors provided concrete examples of sequences that satisfy the new Euler–Gauss rules but fail the older Gauss rules, proving that the new category is strictly larger. They also corrected earlier definitions of related "q-analog" rules, showing that some previously accepted conditions were too weak to guarantee the desired properties. By introducing a modified version of these rules, they created a framework that is both more accurate and more useful for future research. Their work demonstrates that by relaxing a single condition in a famous mathematical rule, one can unlock a rich family of new sequences with their own distinct behaviors and applications.

Ultimately, this research offers a clearer, more detailed view of the relationships between different types of number sequences. It moves beyond the binary question of whether a sequence fits a rule or not, and instead explores the spectrum of possibilities in between. By identifying the Smallest and Greatest Prime Factor sequences as key members of this new family, the authors have highlighted the importance of prime factors in shaping the behavior of these patterns. Their work suggests that the mathematical world is full of subtle variations waiting to be discovered, and that even well-established rules like those of Euler and Gauss have room for expansion. The findings are supported by rigorous proofs and computational checks, giving the mathematical community a solid foundation for exploring these new sequences further.

Technical Summary: On Some Generalisations of Gauss Sequences

Problem Statement
The paper addresses the classification and generalization of integer sequences defined by congruence properties inspired by Euler's theorem ( $a^{\phi(n)} \equiv 1 \pmod n$ ) and Gauss's congruence ( $\sum_{d|n} \mu(d)a^{n/d} \equiv 0 \pmod n$ ). While existing literature extensively covers "Gauss sequences" (also known as Dold sequences) and "Euler sequences," there is a gap in understanding sequences that satisfy a multiplicative analog of the Gauss congruence. Specifically, the authors seek to define a class of sequences that strictly contains Gauss sequences but is strictly contained within Euler sequences, and to explore their $q$ -analogs and connections to the Cyclic Sieving Phenomenon (CSP).

Methodology
The authors employ a combination of number-theoretic definitions, combinatorial analysis, and algebraic manipulation of generating functions.

Definition of Euler–Gauss Sequences: The core methodology involves defining a new congruence condition based on the product of terms indexed by divisors with positive and negative Möbius function values. For a sequence $(a_n)$ , let $A^+_n = \prod_{d|n, \mu(d)=1} a_{n/d}$ and $A^-_n = \prod_{d|n, \mu(d)=-1} a_{n/d}$ . The sequence is defined as an Euler–Gauss sequence if $A^+_n \equiv A^-_n \pmod n$ .
Containment Analysis: The paper rigorously proves the containment hierarchy: $\{\text{Gauss Sequences}\} \subsetneq \{\text{Euler–Gauss Sequences}\} \subsetneq \{\text{Euler Sequences}\}$ . This is achieved by constructing counterexamples (e.g., sequences based on the Smallest and Greatest Prime Factors) that satisfy the Euler–Gauss condition but fail the standard Gauss congruence.
$q$ -Analogs: The authors extend these definitions to polynomial sequences in $\mathbb{Z}[q]$ , introducing $q$ -Euler–Gauss sequences satisfying a congruence modulo the $q$ -integer $[n]_q$ . They analyze the relationship between these and existing $q$ -Gauss sequences, identifying gaps in the literature regarding the equivalence of pairwise congruences in the $q$ -setting.
Divisor Sum Characterizations: The paper investigates sequences of the form $a_n = \sum_{p|n} p f_p(\text{rad}(n))$ , where the sum is over prime divisors, and demonstrates that these naturally form Euler–Gauss sequences.
Cyclic Sieving Phenomenon (CSP): The authors apply the characterization of $q$ -Gauss sequences (specifically the condition $a_n(\omega_n^j) = a_{\gcd(n,j)}(1)$ ) to their new classes to determine CSP triples.

Key Contributions and Results

Introduction of Euler–Gauss Sequences: The paper defines a new class of sequences satisfying the multiplicative congruence $A^+_n \equiv A^-_n \pmod n$ . It proves that every Gauss sequence is an Euler–Gauss sequence, but the converse is false.
Strict Containment and Counterexamples:
- The Smallest Prime Factor (SPF) and Greatest Prime Factor (GPF) sequences (suitably defined at $n=1$ ) are shown to be Euler–Gauss sequences.
- Crucially, the SPF sequence is not a Gauss sequence with respect to any set of primes $S$ , whereas the GPF sequence is a Gauss sequence only with respect to the set $S=\{2\}$ . This establishes that the class of Euler–Gauss sequences is strictly larger than the union of Gauss sequences with respect to any set $S$ .
Strong Euler–Gauss Sequences: A subclass is defined where the ratio $A^+_n / A^-_n$ (in lowest terms) is congruent to $1 \pmod n$ . The paper proves that every strong Euler–Gauss sequence is a Gauss sequence.
$q$ -Analogs and Modified Congruences:
- The authors introduce $q$ -Euler–Gauss sequences.
- They identify that the standard $q$ -analog of the Pairwise Gauss congruence is not equivalent to the $q$ -Gauss congruence. They propose a Modified Pairwise $q$ -Gauss Congruence ( $a_{p^r m}(q) \equiv a_{p^{r-1}m}(q^p) \pmod {[p^r]_{q^m}}$ ) which restores equivalence for $q$ -Gauss sequences.
- Theorem 4.6: A $q$ -Euler–Gauss sequence is a $q$ -Gauss sequence if $a_n(1) \neq 0$ for all $n$ . Consequently, if a $q$ -Euler–Gauss sequence is not a $q$ -Gauss sequence, it must contain at least one zero term at $q=1$ . This implies that strictly Euler–Gauss sequences with all non-zero terms are necessarily $q$ -Gauss sequences.
Divisibility Criterion: The paper conjectures a divisibility criterion for $q$ -Euler–Gauss sequences involving a recursively defined polynomial $A_n(q)$ , verified computationally up to $n=10^8$ .
Cyclic Sieving Phenomenon (CSP):
- The paper extends the CSP results of Gorodetsky and Gossow. It shows that if a $q$ -Euler–Gauss sequence has non-zero terms at $q=1$ for specific divisors, it satisfies the standard CSP condition.
- New CSP Condition: For the $q$ -SPF and $q$ -GPF sequences, the authors derive a novel CSP condition: $|X_n^j| = |X_{\gcd(\text{SPF}(n), j)}|$ . This differs from the standard $|X_n^j| = |X_{\gcd(n, j)}|$ and is shown to be satisfied by the sequences defined in the paper.

Significance and Claims
The paper claims to bridge a fundamental gap between Gauss sequences and Euler sequences by introducing the Euler–Gauss class. The significance lies in:

Unification: Providing a framework that includes sequences like SPF and GPF, which were previously outside the scope of Gauss sequence generalizations.
Clarification of $q$ -Analogs: Addressing gaps in the literature regarding the equivalence of $q$ -congruences and proposing modified definitions (Modified Pairwise $q$ -Gauss) that yield cleaner characterizations.
Combinatorial Interpretation: Demonstrating that these new sequences, particularly the $q$ -SPF and $q$ -GPF, admit combinatorial interpretations via the Cyclic Sieving Phenomenon with a previously unknown fixed-point counting condition.

The authors maintain a modest tone, noting that while they have verified the divisibility criterion computationally, a full proof for the general case remains a conjecture. They emphasize that the classes discussed do not appear in existing literature on integer sequences.

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