Carmichael's Conjecture on the Euler Function is Valid Below 10^{10,000, 000}

1994/07/01 by Aaron Schlafly, Stan Wagon · 16 citations
Mathematics · #Analytic Number Theory Research #History and Theory of Mathematics #Advanced Mathematical Identities #Mathematics #Conjecture #Euler's formula #Function (biology) #Combinatorics #Pure mathematics #Mathematical analysis

paper · doi:10.2307/2153585

published in Mathematics of Computation 63(207), 415 (American Mathematical Society)

openalex publication_date 1994/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11

Abstract

CarmichaeFs conjecture states that if ix) = n , then (y) = n for some y ^ x (^ is Euler's totient function). We show that the conjecture is valid for all x under io10'900'000 . The main new idea is the application of a prime-certification technique that allows us to very quickly certify the primality of the thousands of large numbers that must divide a counterexample.

Citations

• On Euler’s 𝜙-function
• The pseudoprimes to 25⋅10⁹
• Note on Euler’s 𝜑-function
• On a conjecture of Carmichael

Cited by

• Carmichael's totient function conjecture [wikipedia]