The DULA Theorem: Graded Monoid Homomorphisms
for Prime Congruences
DULA - GROK 4
September 19, 2025
Abstract
We present and prove the DULA Theorem, which establishes monoid homomor-
phisms between the multiplicative structure of integers in specific congruence classes
modulo k and the additive structure of (Z/kZ)
. This theorem bridges number theory
and abstract algebra by connecting prime congruence classes with graded algebraic
structures. We provide explicit constructions for various moduli and establish connec-
tions to Dirichlet characters and representation theory.
1 Introduction
The distribution of primes in arithmetic progressions has been a central topic in number
theory since Dirichlet’s seminal work on primes in arithmetic progressions. This paper
establishes a new algebraic framework that connects the multiplicative structure of integers
coprime to a given modulus k with the additive structure of the unit group (Z/kZ)
.
The key insight is that prime factorizations induce natural gradings on monoids of in-
tegers, and these gradings correspond to group homomorphisms that encode congruence
information. This provides a bridge between multiplicative number theory and the repre-
sentation theory of finite abelian groups.
Terminology Note: By a graded monoid, we mean a monoid M equipped with a
homomorphism ϕ : M G to a finite abelian group G, which functions as the grading
group. This differs from the classical notion of graded algebra where one has a direct sum
decomposition M =
L
gG
M
g
. Our grading is instead a homomorphic grading, where the
grade of a product is the sum of the grades of the factors.
2 Preliminaries
Definition 2.1. For a positive integer k and a unit c (Z/kZ)
, define:
P
c
= {p prime : p c (mod k)}
M
c
= {n Z
+
: n c (mod k)}
1
M = {n Z
+
: gcd(n, k) = 1}
Note that M =
S
c(Z/kZ)
M
c
forms a multiplicative monoid generated by primes coprime
to k.
Definition 2.2. For n M with prime factorization n =
Q
p
p
e
p
and a unit c (Z/kZ)
,
define the c-exponent sum:
m
c
(n) =
X
p prime
pc (mod k)
e
p
3 The DULA Theorem: Modulo 6 Case
We begin with the fundamental case of modulo 6, where the structure is most transparent.
Theorem 3.1 (DULA Theorem - Modulo 6). Let M = M
1
M
5
be the multiplicative monoid
of positive integers congruent to 1 or 5 modulo 6. Define the grading function ϕ : M Z/2Z
by
ϕ(n) = m
5
(n) mod 2
and the multiplicative function ψ : M 1} by
ψ(n) = (1)
m
5
(n)
Then:
1. Both ϕ and ψ are monoid homomorphisms
2. There exists an isomorphism θ : Z/2Z 1} such that ψ = θ ϕ
3. The congruence class of n modulo 6 is determined by ϕ(n): n 1 (mod 6) if and only
if ϕ(n) = 0
Proof. Let n M with prime factorization n =
Q
p
p
e
p
.
Step 1: Congruence analysis For primes p ̸∈ {2, 3}:
If p 1 (mod 6), then p
e
p
1 (mod 6)
If p 5 (mod 6), then p
e
p
5
e
p
(mod 6)
Since 5
2
= 25 1 (mod 6), we have:
5
e
p
(
1 (mod 6) if e
p
is even
5 (mod 6) if e
p
is odd
Therefore:
n
Y
p5 (mod 6)
5
e
p
(mod 6)
The product equals 1 modulo 6 if and only if an even number of the exponents e
p
(for
p 5 (mod 6)) are odd, which occurs if and only if m
5
(n) =
P
e
p
is even.
2
Step 2: Homomorphism properties For a, b M:
m
5
(ab) = m
5
(a) + m
5
(b)
since exponents add in prime factorization. Therefore:
ϕ(ab) = m
5
(ab) mod 2 = (m
5
(a) + m
5
(b)) mod 2 = ϕ(a) + ϕ(b) (1)
ψ(ab) = (1)
m
5
(ab)
= (1)
m
5
(a)+m
5
(b)
= (1)
m
5
(a)
· (1)
m
5
(b)
= ψ(a) · ψ(b) (2)
Step 3: Isomorphism Define θ : Z/2Z 1} by θ(0) = 1 and θ(1) = 1. This is a
group isomorphism, and ψ(n) = (1)
m
5
(n)
= θ(m
5
(n) mod 2) = θ(ϕ(n)).
Example 3.2. For n = 25 = 5
2
: m
5
(25) = 2, so ϕ(25) = 0 and ψ(25) = 1. Indeed,
25 1 (mod 6).
For n = 35 = 5 · 7: m
5
(35) = 1, so ϕ(35) = 1 and ψ(35) = 1. Indeed, 35 5
(mod 6).
The following diagram illustrates the relationships in Theorem 3.1:
M 1}
Z/2Z
ψ
ϕ
θ
4 Extension to Modulo 4
Theorem 4.1 (DULA Theorem - Modulo 4). Let M = M
1
M
3
where M
i
= {n i
(mod 4), gcd(n, 4) = 1}. Define ϕ
4
: M Z/2Z by ϕ
4
(n) = m
3
(n) mod 2. Then ϕ
4
is a
monoid homomorphism, and n 1 (mod 4) if and only if ϕ
4
(n) = 0.
Proof. Similar to the modulo 6 case. For primes p odd:
If p 1 (mod 4), then p
e
p
1 (mod 4)
If p 3 (mod 4), then p
e
p
3
e
p
(mod 4)
Since 3
2
1 (mod 4), the congruence class of n modulo 4 is determined by the parity of
m
3
(n).
5 Extension to Prime Power Moduli
Theorem 5.1 (DULA Theorem - Modulo p
k
). Let p be an odd prime and k 1. The group
(Z/p
k
Z)
is cyclic of order ϕ(p
k
) = p
k1
(p 1). Let g be a generator of this group.
For the monoid M of integers coprime to p
k
, there exists a grading function ϕ
p
k
: M
Z(p
k
)Z such that: n g
ϕ
p
k
(n)
(mod p
k
)
3
Proof. Since (Z/p
k
Z)
is cyclic with generator g, every unit u can be written uniquely as
u g
i
(mod p
k
) for some i {0, 1, . . . , ϕ(p
k
) 1}.
For each prime q ̸≡ 0 (mod p), let d
q
be the discrete logarithm of q modulo p
k
with
respect to g. Define: ϕ
p
k
(n) =
P
q
e
q
· d
q
mod ϕ(p
k
) where the sum is over all primes q in
the factorization of n.
The homomorphism property follows from additivity of exponents, and the congruence
relation follows from the definition of discrete logarithm.
Remark 5.2 (Computational Complexity). The construction in this theorem requires com-
puting discrete logarithms modulo p
k
, which is computationally intractable in general for
large primes p. This limits the practical applicability of the grading function for prime power
moduli, though the theoretical framework remains valid. For computational applications,
the theorem is most useful when p is small or when only the existence of the grading is
needed rather than explicit computation.
6 The Universal DULA Theorem
Theorem 6.1 (Universal DULA Theorem). Let k be a positive integer, and let G = (Z/kZ)
.
By the structure theorem for finite abelian groups, there exists an isomorphism
G
=
r
M
i=1
Z/d
i
Z
for some divisors d
i
of ϕ(k).
Let M be the multiplicative monoid of positive integers coprime to k. Then there exists
a grading function
ϕ : M
r
M
i=1
Z/d
i
Z
such that:
1. ϕ is a monoid homomorphism: ϕ(ab) = ϕ(a) + ϕ(b)
2. The congruence class of n modulo k is recovered via: n θ(ϕ(n)) (mod k), where θ is
the isomorphism from the additive to multiplicative form of G
Proof. For each unit c G, let α(c)
L
r
i=1
Z/d
i
Z be its image under the inverse of the
structure isomorphism. For n =
Q
p
p
e
p
, define:
ϕ(n) =
X
p
e
p
· α(p mod k)
The homomorphism property follows from additivity of exponents. The recovery property
follows because:
n mod k =
Y
p
(p mod k)
e
p
= θ
X
p
e
p
· α(p mod k)
!
= θ(ϕ(n))
4
7 Connection to Dirichlet Characters
The DULA theorem has deep connections to the theory of Dirichlet characters.
Theorem 7.1 (DULA and Dirichlet Characters). Let χ be a Dirichlet character modulo k.
Then for any n M:
χ(n) = χ
add
(ϕ(n))
where χ
add
is the corresponding additive character on
L
r
i=1
Z/d
i
Z and ϕ is the DULA grading
function.
Proof. Since χ is a group homomorphism from (Z/kZ)
to C
, and ϕ provides the additive
representation of the group structure, we have:
χ(n) = χ(n mod k) = χ(θ(ϕ(n))) = (χ θ)(ϕ(n))
where χ θ is an additive character.
8 Extension to Non-Cyclic Cases: Modulo 8
The modulo 8 case illustrates how the DULA framework handles non-cyclic unit groups.
Theorem 8.1 (DULA Theorem - Modulo 8). Let M =
S
i∈{1,3,5,7}
M
i
where M
i
= {n i
(mod 8), gcd(n, 8) = 1}.
The group (Z/8Z)
= {1, 3, 5, 7}
=
Z/2Z × Z/2Z is the Klein four-group with generators
3 and 5.
Define the grading function ϕ
8
: M Z/2Z×Z/2Z by: ϕ
8
(n) = (m
3
(n) mod 2, m
5
(n) mod
2) where m
3
(n) =
P
p3 (mod 8)
e
p
and m
5
(n) =
P
p5 (mod 8)
e
p
.
Then ϕ
8
is a monoid homomorphism, and the congruence class of n modulo 8 is deter-
mined by:
(0, 0) n 1 (mod 8) (3)
(1, 0) n 3 (mod 8) (4)
(0, 1) n 5 (mod 8) (5)
(1, 1) n 7 (mod 8) (6)
Proof. The key observation is that (Z/8Z)
has the structure:
1 = 3
0
· 5
0
(identity)
3 = 3
1
· 5
0
(generator)
5 = 3
0
· 5
1
(generator)
7 = 3
1
· 5
1
(product of generators)
Note that 7 1 (mod 8), and indeed 7 = 3 · 5 8 = 15 8.
For primes p > 2:
5
If p 1 (mod 8): contributes (0, 0) to the grading
If p 3 (mod 8): contributes (1, 0) per unit exponent
If p 5 (mod 8): contributes (0, 1) per unit exponent
If p 7 (mod 8): since 7 3 · 5 (mod 8), this is handled by the existing generators
The homomorphism property follows from the additivity of exponents, and the congru-
ence correspondence follows from the group structure of (Z/8Z)
.
Example 8.2 (Modulo 8 Computation). Consider n = 15 = 3 · 5:
m
3
(15) = 1 (one factor of 3)
m
5
(15) = 1 (one factor of 5)
ϕ
8
(15) = (1, 1)
Therefore 15 7 (mod 8)
Consider n = 45 = 3
2
· 5:
m
3
(45) = 2 (exponent 2 for prime 3)
m
5
(45) = 1 (exponent 1 for prime 5)
ϕ
8
(45) = (0, 1)
Therefore 45 5 (mod 8)
This example demonstrates how the DULA framework naturally handles the Klein four-
group structure, with each component of the grading corresponding to one of the independent
generators of the group.
9 Applications and Future Directions
9.1 Connection to L-functions and Prime Distribution
The DULA theorem provides new tools for analyzing L-functions and prime distributions.
For a Dirichlet character χ modulo k, the Dirichlet L-function is:
L(s, χ) =
X
n=1
χ(n)
n
s
=
Y
p
1
χ(p)
p
s
1
Using the DULA grading ϕ, we can rewrite this as:
L(s, χ) =
Y
p
1
χ
add
(ϕ(p))
p
s
1
This reformulation allows us to:
6
Group primes by their grading values rather than congruence classes
Analyze the distribution of gradings among primes
Connect multiplicative properties of L-functions to additive structures
9.2 Prime Constellations and Twin Primes
The DULA framework naturally extends to studying prime constellations. For twin primes
(p, p + 2), both must avoid certain congruence classes. The grading functions can help
characterize admissible patterns.
Proposition 9.1 (Twin Prime Constraint). For twin primes (p, p + 2) with p > 3, if p 5
(mod 6), then p + 2 1 (mod 6). In terms of DULA gradings modulo 6:
ϕ
6
(p) = 1 (contributing to the odd class)
ϕ
6
(p + 2) = 0 (contributing to the even class)
This asymmetry in gradings reflects the inherent structure of twin prime pairs.
9.3 Computational Applications
Character evaluation: For small moduli where discrete logs are feasible, DULA
gradings provide efficient character evaluation
Factorization patterns: The grading functions can detect certain factorization pat-
terns without full prime decomposition
Pseudoprimality testing: Grading-based tests could supplement existing primality
criteria
9.4 Open Problems and Extensions
1. Quantitative density results: Can DULA gradings provide new bounds on the
density of primes in arithmetic progressions?
2. Higher-dimensional analogues: Extend the framework to number fields and their
units groups
3. Effective bounds: For which moduli k can the discrete logarithms in the grading
functions be computed efficiently?
4. Connection to class field theory: How do DULA gradings relate to Artin reci-
procity and class field towers?
5. Analytic number theory: Can the additive structure revealed by DULA gradings
lead to new estimates for error terms in the prime number theorem for arithmetic
progressions?
7
10 Conclusion
The DULA theorem establishes a fundamental connection between the multiplicative struc-
ture of integers and the additive structure of unit groups modulo k. This bridge between
number theory and abstract algebra opens new avenues for research in both fields.
The universal formulation shows that this phenomenon is not isolated to specific moduli
but reflects deeper structural properties of finite abelian groups. The connections to Dirichlet
characters and representation theory suggest rich possibilities for future investigation.
References
[1] T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.
[2] H. Davenport, Multiplicative Number Theory, 3rd ed., Graduate Texts in Mathematics,
vol. 74, Springer, 2000.
[3] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, 2nd ed.,
Graduate Texts in Mathematics, vol. 84, Springer, 1990.
[4] J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics,
vol. 42, Springer, 1977.
[5] L. C. Washington, Introduction to Cyclotomic Fields, 2nd ed., Graduate Texts in Math-
ematics, vol. 83, Springer, 1997.
8