# Carmichael's totient function conjecture

In mathematics,

**Carmichael's totient function conjecture**concerns themultiplicity of values ofEuler's totient function φ("n"), which counts the number of integers less than andcoprime to "n".This function φ("n") is equal to 2 when "n" is one of the three values 3, 4, and 6. It is equal to 4 when "n" is one of the four values 5, 8, 10, and 12. It is equal to 6 when "n" is one of the four values 7, 9, 14, and 18. In each case, there is more than one value of "n" having the same value of φ("n").

The conjecture states that this phenomenon of repeated values holds for every "n". That is, for every "n" there is at least one other integer "m" ≠ "n" such that φ("m") = φ("n").

Robert Carmichael first stated this conjecture 1907, but as a theorem rather than as a conjecture. However, his proof was faulty and in 1922 he retracted his claim and stated the conjecture as an open problem.Carmichael proved that any counterexample to his conjecture (that is, a value "n" such that φ("n") is different from the totients of all other numbers) must be at least 10^{37}, andVictor Klee extended this result to 10^{400}. Carmichael's conjecture has since been verified computationally for all "n" less than or equal to 10^{107}by Schlafly and Wagon. The current lower bound for a counterexample to the Conjecture is 10^{1010}, which was determined by Kevin Ford in 1998. Ford went on to prove that if there exists a counterexample to the Conjecture, then a positive fraction (that is infinitely many) of the integers are likewise counterexamples.Carmichael's Conjecture is one of the first among open problems with very high lower bounds and which are relatively easy to determine. The computational technique basically depends on some key results of Klee which makes it possible to show that the smallest counterexample must be divisible by squares of the primes dividing it. Klee's results imply that 8 and Fermat primes (primes of the form 2

^{"k"}+1) excluding 3 do not divide the smallest counterexample. Consequently, proving the conjecture is equivalent to proving that the conjecture holds for all integers congruent to 4 ("mod" 8).Although the conjecture is widely believed,

Carl Pomerance gave a sufficient condition for an integer "n" to be a counterexample to the conjecture. According to this condition, "n" is a counterexample if for every prime "p" such that "p" − 1 divides φ("n"), "p"^{2}divides "n". However Pomerance showed that the existence of such an integer is highly improbable. Essentially, one can show that if the first "k" primes "p" congruent to 1 ("mod q") (where "q" is a prime) are all less than "q"^{"k"+1}, then such an integer will be divisible by every prime and thus cannot exist. In any case, proving that Pomerance's counterexample does not exist is far from proving Carmichael's Conjecture. However if it exists then infinitely many counterexamples exist as asserted by Ford.Another way of stating Carmichael's conjecture is that, ifA("f") denotes the number of positive integers "n" for which φ("n") = "f", then A("f") can never equal 1. Relatedly,

Wacław Sierpiński conjectured that every positive integer other than 1 occurs as a value of A("f"), a conjecture that was proven in 1999 by Kevin Ford.**References***citation

last = Carmichael | first = R. D. | author-link = Robert Daniel Carmichael

doi = 10.1090/S0002-9904-1907-01453-2

id = MR|1558451

issue = 5

journal =Bulletin of the American Mathematical Society

pages = 241–243

title = On Euler's φ-function

volume = 13

year = 1907.*citation

last = Carmichael | first = R. D. | author-link = Robert Daniel Carmichael

doi = 10.1090/S0002-9904-1922-03504-5

id = MR|1560520

issue = 3

journal =Bulletin of the American Mathematical Society

pages = 109–110

title = Note on Euler's φ-function

volume = 28

year = 1922.*citation

last = Ford | first = K.

doi = 10.2307/121103

id = MR|1715326

issue = 1

journal =Annals of Mathematics

pages = 283–311

title = The number of solutions of φ("x") = "m"

volume = 150

year = 1999.*citation

last = Klee | first = V. L., Jr. | author-link = Victor Klee

doi = 10.1090/S0002-9904-1947-08940-0

id = MR|0022855

journal =Bulletin of the American Mathematical Society

pages = 1183–1186

title = On a conjecture of Carmichael

volume = 53

year = 1947.*citation

last1 = Schlafly | first1 = A.

last2 = Wagon | first2 = S.

doi = 10.2307/2153585

id = MR|1226815

issue = 207

journal =Mathematics of Computation

pages = 415–419

title = Carmichael's conjecture on the Euler function is valid below 10^{10,000,000}

volume = 63

year = 1994.**External links***mathworld|title=Carmichael's Totient Function Conjecture|urlname=CarmichaelsTotientFunctionConjecture

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