Using calculus to do number theory
2 days ago
- #Modular Arithmetic
- #Hensel's Lemma
- #Number Theory
- Calculus and number theory, seemingly unrelated, are connected through Kurt Hensel's discovery.
- Hensel's method uses calculus to solve polynomial equations in modular arithmetic, exemplified by solving \(x^3 - 17x^2 + 12x + 16 \equiv 0 \pmod{3000}\).
- The Chinese Remainder Theorem breaks the problem into simpler congruences modulo prime powers.
- Newton's method from calculus is adapted to improve solutions modulo higher powers of primes.
- Hensel's lemma reduces solving \(f(x) \equiv 0 \pmod{p^e}\) to solving \(f(x) \equiv 0 \pmod{p}\).
- The problem of solving polynomial congruences leads to deep mathematics, including the Langlands program.
- Class field theory addresses solutions for polynomials with abelian Galois groups, while non-abelian cases are more complex.