Fermat's Library | A Combinatorial Approach to Identifying Primes via Congruence Classes Modulo 6 annotated/explained version.

A Combinatorial Approach to Identifying Primes via

Congruence Classes Modulo 6

DULA

GROK 3

February 22, 2025

Abstract

We present a novel method to systematically identify all prime numbers greater

than 3 by utilizing the arithmetic structure of congruence classes modulo 6. By

ﬁrst eliminating numbers divisible by 2 or 3, we restrict our attention to the classes

1 and 5 modulo 6, which contain all primes greater than 3. We then employ

a combinatorial strategy, generating all possible products of primes within these

classes—including powers and multiple factors—to exhaustively eliminate compos-

ites. The remaining numbers are precisely the primes. This approach leverages

closure properties under multiplication and provides a proof by exhaustion, of-

fering an alternative perspective on prime identiﬁcation. Examples and algebraic

insights, including a graded monoid structure, enhance the exposition.

1 Introduction

Prime numbers, the building blocks of the integers, have fascinated mathematicians for

centuries. Traditional methods like the Sieve of Eratosthenes eﬃciently identify primes by

eliminating multiples of known primes. In this paper, we explore a combinatorial method

that exploits the residue classes modulo 6 to isolate primes greater than 3. Since 6 = 2

× 3, numbers coprime to 6 are not divisible by 2 or 3, restricting primes greater than 3

to the classes 1 and 5 modulo 6. We then systematically generate all composites within

these classes using multiplication and exponentiation, leaving only the primes behind.

This method not only identiﬁes primes but also reveals an algebraic structure—a

graded monoid—arising from the multiplicative properties of these numbers. We aim

to make this accessible with concrete examples and rigorous proofs, contributing a fresh

approach to number theory.

2 Preliminaries

Consider the integers modulo 6, with residue classes {0, 1, 2, 3, 4, 5}. A prime number

p > 3 cannot be divisible by 2 or 3, the prime factors of 6. Analyzing each class:

• 0 (mod 6): e.g., 6, 12, 18—all divisible by 6, hence composite.

• 2 (mod 6): e.g., 2, 8, 14—divisible by 2; only 2 is prime, excluded since p > 3.

• 3 (mod 6): e.g., 3, 9, 15—divisible by 3; only 3 is prime, excluded.

• 4 (mod 6): e.g., 4, 10, 16—all divisible by 2, composite.

• 1 (mod 6): e.g., 7, 13, 19—potential primes.

• 5 (mod 6): e.g., 5, 11, 17—potential primes.

Thus, all primes p > 3 are congruent to 1 or 5 modulo 6.

Deﬁne: - P

= {p prime : p > 3, p ≡ 1 (mod 6)}, e.g., {7, 13, 19, . . . }, - P

{p prime : p > 3, p ≡ 5 (mod 6)}, e.g., {5, 11, 17, . . . }, - P = P

∪ P

, - S = {n ∈ Z

n =

p∈P

, e

≥ 0}, the multiplicative monoid generated by P (including 1 when all

= 0).

Every n ∈ S is congruent to 1 or 5 modulo 6, as products of elements in P remain in

these classes.

3 Multiplicative Closure and Algebraic Structure

The set S is closed under multiplication:

• 1 · 1 = 1 (mod 6), e.g., 7 · 13 = 91 ≡ 1,

• 1 · 5 = 5 (mod 6), e.g., 7 · 5 = 35 ≡ 5,

• 5 · 5 = 25 ≡ 1 (mod 6), e.g., 5 · 11 = 55 ≡ 5, 11 · 11 = 121 ≡ 1.

Partition S = S

∪ S

, where: - S

= {n ∈ S : n ≡ 1 (mod 6)}, - S

= {n ∈ S : n ≡ 5

(mod 6)}.

For n =

, let m =

≡5

(sum of exponents of primes 5 mod 6). Since 5 ≡ −1

(mod 6) and 1 ≡ 1 (mod 6),

n ≡

(1)

(5)

≡ 1 · (−1)

≡ (−1)

(mod 6).

- m even: n ≡ 1 (mod 6), e.g., 5

· 11 = 275, m = 2 + 1 = 3 odd, but 5 · 5 = 25 ≡ 1,

25 · 11 = 275 ≡ 5, - m odd: n ≡ 5 (mod 6).

This deﬁnes a Z/2Z-grading: degree 0 for S

, degree 1 for S

4 Combinatorial Elimination of Composites

4.1 Stage 1: Restricting to S

By excluding numbers divisible by 2 or 3, S contains all n ≡ 1 or 5 (mod 6), including

primes and composites (e.g., 25 = 5

, 35 = 5 × 7).

4.2 Stage 2: Exhausting Composites

Generate all composites in S as products of two or more factors from P : - 5

= 25 ≡ 1,

- 5 · 7 = 35 ≡ 5, - 5 · 7 · 11 = 385 ≡ 1 (5 · 5 = 25 ≡ 1, 25 · 11 = 385), - 5

· 7

· 11 =

25 · 343 · 11 = 94325 ≡ 5 (25 ≡ 1, 343 ≡ 1, 11 ≡ 5).

Up to 50, list S: 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49. Composites

include: - 25 = 5

, - 35 = 5 × 7, - 49 = 7

Remaining: 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47—all primes.

4.3 Proof

Every composite in S has at least two prime factors from P . Generating all such products

exhausts all composites. Numbers in S with exactly one prime factor (exponent sum =

1) are the primes in P .

5 Conclusion

This combinatorial method, while less eﬃcient than sieving, provides a systematic proof

by exhaustion for identifying primes greater than 3. It highlights the interplay of arith-

metic modulo 6 and algebraic structures, oﬀering educational value and a fresh perspective

on prime number theory.

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