Carmichael's totient function conjecture, in number theory, concerns the multiplicity of values of Euler's totient function. It asserts that for every n there is at least one other integer m, not equal to n, such that the totient of m equals the totient of n. Robert Carmichael first presented the claim in 1907 as a theorem, but his proof contained errors, and he retracted it in 1922, restating it as an open conjecture rather than a proven result.
Facts
StatementFor every n there is at least one other integer m not equal to n such that the totient of m equals the totient of n. 1 Proposed Year Progress Toward ResolutionOpen; any counterexample must exceed 10^(10^10), a bound determined by Kevin Ford in 1998. 1 Difficulty Hint Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Wolfram MathWorld: Carmichael's Totient Function
Conjecture
Posed By
Source Wikipedia: Carmichael Totient Function Conjecture
Sources
1. Carmichael's Totient Function Conjecture (Wikipedia)
Wikimedia FoundationLead section
Robert Carmichael first stated this conjecture in 1907, but as a theorem rather than as a conjecture.
Lead section, first sentence
concerns the multiplicity of values of Euler's totient function
Lower bounds
was determined by Kevin Ford in 1998
View the Source 2. Wikipedia: Carmichael Totient Function Conjecture
WikipediaLead section
concerns the multiplicity of values of Euler
Lead section, resolution status
However, his proof was faulty, and in 1922, he retracted his claim and stated the conjecture as an open problem.
View the Source Wolfram MathWorld: Carmichael's Totient Function
Conjecture
Wolfram MathWorldIn Branch: Number Theory, Subject classificationsQuote, In Branch: Number Theory, Subject classifications
Number Theory Number Theoretic Functions
Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.