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Enhancing the Frobenius Primality Test Using Cubic Recurrence Relations

Introduction

In the pursuit of developing robust primality tests, the integration of symmetric polynomials, linear recurrence relations, and the Frobenius automorphism within the framework of Galois Theory offers a profound and efficient methodology. Building upon the foundational understanding of how the Frobenius automorphism cyclically permutes the roots of an irreducible cubic polynomial, we explore the derivation of sequence terms essential for the primality test. This document corrects and finalizes the expression for ( V_{p+2} ), ensuring the test’s reliability and reducing the likelihood of pseudoprimes passing the test.

Recurrence Relation for Cubic Polynomials

Consider the irreducible cubic polynomial over the finite field ( \mathbb{F}p ): $$ f(x) = x^3 + Sx^2 + Px + Q = 0 $$ with distinct roots ( \alpha ), ( \beta ), ( \gamma ) in the extension field ( \mathbb{F}{p^3} ).

Defining the Sequence ( V_k )

We define the sequence ( V_k ) as the sum of the ( k )-th powers of the roots:
$$
V_k = \alpha^k + \beta^k + \gamma^k \quad \text{for } k = 0, 1, 2, \dots
$$
From Vieta’s formulas, we have:
$$
\alpha + \beta + \gamma = -S, \quad \alpha\beta + \alpha\gamma + \beta\gamma = P, \quad \alpha\beta\gamma = -Q
$$

Establishing the Correct Recurrence Relation

Given the minimal polynomial ( f(x) = 0 ), each root satisfies:
$$
\alpha^3 = -S\alpha^2 – P\alpha – Q
$$
$$
\beta^3 = -S\beta^2 – P\beta – Q
$$
$$
\gamma^3 = -S\gamma^2 – P\gamma – Q
$$

Multiplying each equation by ( \alpha^{k-3} ), ( \beta^{k-3} ), ( \gamma^{k-3} ) respectively and summing, we obtain the linear recurrence relation:
$$
V_k = -S V_{k-1} – P V_{k-2} – Q V_{k-3} \quad \text{for } k \geq 3
$$

Frobenius Automorphism and Root Permutation

Cyclic Permutation of Roots

In the context of finite fields, the Frobenius automorphism ( \sigma ) maps each root to its ( p )-th power:
$$
\sigma(\alpha) = \alpha^p = \beta
$$
$$
\sigma(\beta) = \beta^p = \gamma
$$
$$
\sigma(\gamma) = \gamma^p = \alpha
$$
This cyclic permutation arises because ( f(x) ) is irreducible over ( \mathbb{F}_p ), ensuring that the Frobenius automorphism generates the entire Galois group, which is cyclic of order 3 in this cubic case.

Determining Special Values ( V_p ), ( V_{p+1} ), and ( V_{p+2} )

Value of ( V_p )

$$
V_p = \alpha^p + \beta^p + \gamma^p = \beta + \gamma + \alpha = \alpha + \beta + \gamma = -S
$$
Result:
$$
V_p = -S
$$

Value of ( V_{p+1} )

$$
V_{p+1} = \alpha^{p+1} + \beta^{p+1} + \gamma^{p+1} = \alpha\beta + \beta\gamma + \gamma\alpha = P
$$
Result:
$$
V_{p+1} = P
$$

Value of ( V_{p+2} )

To derive ( V_{p+2} ), we proceed as follows:

$$
V_{p+2} = \alpha^{p+2} + \beta^{p+2} + \gamma^{p+2} = \alpha^p \cdot \alpha^2 + \beta^p \cdot \beta^2 + \gamma^p \cdot \gamma^2 = \beta \alpha^2 + \gamma \beta^2 + \alpha \gamma^2
$$

Expressing in Terms of Symmetric Polynomials

We aim to express ( V_{p+2} = \beta \alpha^2 + \gamma \beta^2 + \alpha \gamma^2 ) using symmetric polynomials. Consider the following symmetric sums:

1. Sum of All Cyclic Permutations:
   $$
   \alpha^2 \beta + \beta^2 \gamma + \gamma^2 \alpha
   $$
2. Sum of All Anticyclic Permutations:
   $$
   \alpha \beta^2 + \beta \gamma^2 + \gamma \alpha^2
   $$
3. Total Symmetric Sum:
   $$
   \alpha^2 \beta + \beta^2 \gamma + \gamma^2 \alpha + \alpha \beta^2 + \beta \gamma^2 + \gamma \alpha^2
   $$

From symmetric polynomial theory, we have the identity:
$$
\alpha^2 \beta + \beta^2 \gamma + \gamma^2 \alpha + \alpha \beta^2 + \beta \gamma^2 + \gamma \alpha^2 = \sigma_1 \sigma_2 – 3\sigma_3
$$
where:
$$
\sigma_1 = \alpha + \beta + \gamma = -S
$$
$$
\sigma_2 = \alpha\beta + \alpha\gamma + \beta\gamma = P
$$
$$
\sigma_3 = \alpha\beta\gamma = -Q
$$
Substituting these values:
$$
\alpha^2 \beta + \beta^2 \gamma + \gamma^2 \alpha + \alpha \beta^2 + \beta \gamma^2 + \gamma \alpha^2 = (-S)(P) – 3(-Q) = -SP + 3Q
$$
Thus:
$$
\alpha^2 \beta + \beta^2 \gamma + \gamma^2 \alpha + \alpha \beta^2 + \beta \gamma^2 + \gamma \alpha^2 = -SP + 3Q
$$`

Relating to ( V_{p+2} ):

Given:
$$
V_{p+2} = \beta \alpha^2 + \gamma \beta^2 + \alpha \gamma^2 = \alpha^2 \beta + \beta^2 \gamma + \gamma^2 \alpha
$$
Let:
$$
A = \alpha^2 \beta + \beta^2 \gamma + \gamma^2 \alpha
$$
$$
B = \alpha \beta^2 + \beta \gamma^2 + \gamma \alpha^2
$$
We have:
$$
A + B = -SP + 3Q
$$
Therefore:
$$
A = \frac{ -SP + 3Q }{ 2 }
$$
Result:
$$
V_{p+2} = A = \frac{ -S P + 3 Q }{ 2 }
$$

Verification with an Example

Let’s verify this expression with your provided example.

Example:
$$
f(x) = x^3 – x – 1 = 0 \quad \Rightarrow \quad S = 0, \ P = -1, \ Q = -1
$$
Substituting into the expression for ( V_{p+2} ):
$$
V_{p+2} = \frac{ -0 \times (-1) + 3 \times (-1) }{ 2 } = \frac{ 0 – 3 }{ 2 } = -\frac{3}{2}
$$
Thus,
$$
V_{p+2} = -\frac{3}{2}
$$
Your PARI/gp computation confirms this result:
$$
(-P \cdot S + 3Q)/2 – V_{p+2} \approx 0
$$
indicating that:
$$
V_{p+2} = \frac{ -S P + 3 Q }{ 2 }
$$
is indeed correct.

Primality Testing Conditions

For a candidate number ( n = p ) to be prime, the following congruence conditions must simultaneously hold:
$$
V_p \equiv -S \pmod{p}
$$
$$
V_{p+1} \equiv P \pmod{p}
$$
$$
V_{p+2} \equiv \frac{ -S P + 3 Q }{ 2 } \pmod{p}
$$
These conditions are derived from the properties of the Frobenius automorphism and the established recurrence relations.

Matrix Representation of the Recurrence Relation

To facilitate efficient computation and analysis, the cubic recurrence relation can be elegantly represented in matrix form. Define the state vector as:
$$
\mathbf{S}k = \begin{bmatrix} V{k} \ V_{k+1} \ V_{k+2} \end{bmatrix}
$$
The transition matrix ( \mathbf{T} ) is then:
$$
\mathbf{T} = \begin{bmatrix}
0 & 1 & 0 \
0 & 0 & 1 \

• Q & – P & – S \
  \end{bmatrix}
  $$
  This matrix encapsulates the recurrence relation:
  $$
  V_k = -S V_{k-1} – P V_{k-2} – Q V_{k-3}
  $$
  Ensuring that ( \mathbf{T} ) is invertible is crucial for the test’s strength, as it guarantees that the sequence ( V_k ) evolves deterministically without degeneracies.

Conclusion

Through meticulous derivation and verification using PARI/gp, we’ve confirmed that:
$$
V_{p+2} = \frac{ -S P + 3 Q }{ 2 }
$$
This expression is pivotal for the Cubic Frobenius Primality Test, providing essential congruence conditions that a prime number must satisfy relative to the chosen cubic polynomial’s coefficients. By verifying these conditions, especially across multiple polynomials, the test gains robustness and reliability in distinguishing primes from composites.

Key Takeaways:

• Accurate Derivations: Ensuring correct application of the recurrence relation and Frobenius automorphism is crucial for establishing valid primality conditions.
• Symmetric Polynomials: Utilizing symmetric polynomial identities allows for expressing higher-order sequence terms in terms of polynomial coefficients.
• Matrix Representation Benefits: Facilitates efficient computation and provides a foundation for deeper analytical insights into the sequence’s behavior.
• Empirical Verification: Confirming theoretical derivations with computational tools like PARI/gp enhances confidence in the framework’s accuracy.

Further Reading

1. Abstract Algebra by David S. Dummit and Richard M. Foote
2. Galois Theory by Ian Stewart
3. Finite Fields by Rudolf Lidl and Harald Niederreiter

Stay Engaged and Continue Exploring the Elegant Synergies of Algebra and Number Theory! 😊

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