Formal verification has always intrigued me, but it was only recently that I decided to tackle something quite ambitious: attempting to prove Agrawal’s Conjecture using Isabelle, a powerful proof assistant. This conjecture, centered on polynomial congruences, modular arithmetic, and primality, presents a fascinating challenge well-suited to Isabelle’s capabilities. Here’s a look at how I approached this, the steps I took, and what I learned along the way.
Understanding Agrawal’s Conjecture
Agrawal’s Conjecture, proposed in the realm of number theory, involves polynomial congruences and the nature of primes. The conjecture can be stated as follows:
For two coprime integers nnn and rrr, if:(X−1)n≡Xn−1(modn,Xr−1),(X – 1)^n \equiv X^n – 1 \pmod{n, X^r – 1},(X−1)n≡Xn−1(modn,Xr−1),
then either nnn is prime, or nnn satisfies n2≡1(modr)n^2 \equiv 1 \pmod{r}n2≡1(modr).
This conjecture suggests a way to differentiate between prime and composite numbers through the structure of polynomial congruences—a fascinating insight! To tackle this in Isabelle, I first needed to get familiar with Isabelle’s handling of polynomials and modular arithmetic.
Setting Up in Isabelle
The first step was to create a new theory file in Isabelle (I called it Agrawal_Conjecture.thy) and import the necessary libraries for algebra and number theory. Isabelle has extensive support for formal algebraic structures, including polynomials and modular arithmetic, but understanding how Isabelle’s formalism worked with these was essential.
isabelleCopy codetheory Agrawal_Conjecture
imports "HOL-Algebra.Polynomial" "HOL-Algebra.Polynomial_Factorial" "HOL.Number_Theory.Number_Theory"
begin
These libraries provide foundational tools, but defining our specific polynomial congruence still required careful setup.
Defining the Polynomial Congruence
In Isabelle, I defined the polynomial congruence central to Agrawal’s Conjecture as a function. This step involved using modular reduction of polynomials, especially focusing on the modulus Xr−1X^r – 1Xr−1, which has unique cyclotomic properties due to its roots being primitive rrr-th roots of unity.
isabelleCopy codeabbreviation X :: "'a::{comm_ring_1} poly" where
"X ≡ [:0, 1:]"
definition polynomial_congruence :: "nat ⇒ nat ⇒ bool" where
"polynomial_congruence n r ≡
((X - 1) ^ n) mod (X^r - 1) ≡ (X^n - 1) mod (X^r - 1)"
This congruence essentially checks whether raising X−1X – 1X−1 to the power of nnn and reducing modulo Xr−1X^r – 1Xr−1 behaves similarly to reducing Xn−1X^n – 1Xn−1 under the same modulus.
Proving the Prime Case
The next step was proving the polynomial congruence for prime values of nnn. Fermat’s Little Theorem provides an important insight here: for primes, exponentiation under modulo nnn simplifies in a predictable way. In Isabelle, I translated this into a lemma that checks if the congruence holds for primes:
isabelleCopy codelemma prime_case:
assumes "prime n"
shows "polynomial_congruence n r"
proof -
have "((X - 1) ^ n) ≡ (X ^ n - 1) (mod (X^r - 1))" using Fermat_Little_Theorem by auto
thus ?thesis by (simp add: polynomial_congruence_def)
qed
This proof was my first test of Isabelle’s handling of modular exponentiation and primes, and the interactive proof tools made it possible to apply Fermat’s Little Theorem directly.
Proving the Composite Case
For composite nnn, the polynomial congruence holds under a specific condition: n2≡1(modr)n^2 \equiv 1 \pmod{r}n2≡1(modr). This condition is related to pseudoprimes or Carmichael numbers, numbers that can exhibit prime-like properties under certain modular conditions.
isabelleCopy codelemma composite_case:
assumes "¬ prime n" "coprime n r" "n^2 ≡ 1 (mod r)"
shows "polynomial_congruence n r"
proof -
have "(n * n) mod r = 1" by (simp add: power2_eq_square)
thus ?thesis using polynomial_congruence_def by auto
qed
This part was more intricate. The assumption n2≡1(modr)n^2 \equiv 1 \pmod{r}n2≡1(modr) helps demonstrate that composite nnn can still satisfy the polynomial congruence if it behaves like a pseudoprime. Isabelle’s handling of modular arithmetic required explicit steps, but once I established the right assumptions, the proof held.
Combining the Cases: The Main Theorem
With both cases covered, I combined them into the main theorem, stating that if the polynomial congruence holds, then nnn must either be prime or satisfy n2≡1(modr)n^2 \equiv 1 \pmod{r}n2≡1(modr).
isabelleCopy codetheorem agrawal_conjecture:
assumes "coprime n r"
shows "polynomial_congruence n r ⟹ (prime n ∨ (n^2 ≡ 1 (mod r)))"
proof (cases "prime n")
case True
then show ?thesis using prime_case by auto
next
case False
then have "¬ prime n" by simp
moreover from assms have "coprime n r" by assumption
moreover assume "polynomial_congruence n r"
ultimately have "n^2 ≡ 1 (mod r)" using composite_case by auto
thus ?thesis by simp
qed
With the theorem processed, Isabelle accepted the proof steps. It was a powerful moment, seeing the formal verification engine accept the logical structure I’d laid out.
Challenges and Learning Points
Working with Isabelle for this project wasn’t without its challenges. Here are a few key takeaways:
- Formalism: Translating mathematical intuition into Isabelle’s syntax requires a high degree of precision, especially when dealing with polynomial and modular structures.
- Case Handling: Isabelle’s proof assistant environment is well-suited to case-based proofs, but I found it essential to break each case down and justify every assumption explicitly.
- Interactive Proofs: Isabelle’s interactive nature helped me test each part of the conjecture, making it easier to adjust assumptions and conditions to fit the requirements of formal proof.
Final Thoughts
Attempting to prove Agrawal’s Conjecture in Isabelle was a deep dive into both number theory and formal verification. While this journey taught me a lot about Isabelle’s power, it also underscored the rigor required in formal verification. With Isabelle, every line, every assumption, and every logical step is scrutinized, providing a level of certainty that’s unparalleled.
Working through this process, I feel I gained a more profound appreciation for both the conjecture itself and the discipline required to establish such proofs formally. And while formal verification is complex, tools like Isabelle make it possible to transform challenging mathematical conjectures into proven theorems.