Pierre de Fermat's Link to a High School Student's Prime Math Proof | Quanta Magazine

Abstraction: Carmichael numbers as pseudoprime counterexamples to Fermat's little theorem

Key points:

• Fermat's little theorem: for prime p and any integer a, a^p - a is divisible by p
• Carmichael numbers are composite numbers that satisfy Fermat's little theorem for all a; first example is 561 = 3 x 11 x 17
• High school student Daniel Larsen (2022) proved Carmichael numbers are more densely distributed than previously shown, analogous to Bertrand's postulate for primes
• Larsen built on work of Fields medalists James Maynard and Terence Tao
• The converse of Fermat's little theorem is false, making primality testing via the theorem unreliable alone
• Carmichael numbers named after mathematician Robert Carmichael; infinitely many exist

Connections: Number Theory · Prime Numbers

Source: https://www.quantamagazine.org/pierre-de-fermats-link-to-a-high-school-students-prime-math-proof-20231122/