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Carmichael's Totient Conjecture
.. Third: Yes! Carmichael's totient function conjecture is a real world unsolved math proof (untill 1999) first stated by American mathematician Robert Daniel Carmichael as a theorem rather than a conjecture in 1907. The University of Illinois mathematics professor - who was the President of Mathematical Association of America in 1923 - stated that: for every n there is at least one other integer m ≠ n such that φ(m) = φ(n). However, his proof was faulty and in 1922 he retracted his claim and stated the conjecture as an open problem (In Science & Math an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved - no solution for it is known). The conjecture was finally proven in 1999 ( Kevin Ford Annals of Mathematics Second Series, Vol. 150, No. 1 (Jul., 1999), pp. 283-311 )

Conjecture:
Noun
  • (formal) A statement or an idea which is unproven, but is thought to be true; a .
I explained it, but it is pure conjecture whether he understood, or not.
  • (formal) A supposition based upon incomplete evidence; a hypothesis.
The physicist used his conjecture about subatomic particles to design an experiment.
  • (mathematics, philology) A statement likely to be true based on available evidence, but which has not been formally (l).
  • (obsolete) of signs and omens.

Synonyms
* * See also
Related terms
* conject * conjecturalVerb(conjectur)
  • (formal) To ; to venture an unproven idea.
I do not know if it is true; I am simply conjecturing here.
  • * South
Human reason can then, at the best, but conjecture what will be

Robert Carmichael proposed one such conundrum in 1907 that still remains unsolved.   Basically, Carmichael conjectured that for every positive integer there exists a positive integer "m" such that m ≠ n and  φ(m) = φ(n).  

As a consequence, with the given properties, the conjecture is certainly true for odd numbers.  This can be seen by letting n be a positive odd integer and in the fact that  φ(2) =1

φ(2)= φ(2) φ(n) =φ(n)


However,  as easily proved as the conjecture is for the positive odd integers,  the statement has not been shown true for the positive even integers.
Proof:

Suppose n is prime and n−1= k

Then φ(n)=k
Now φ(2)=1,

and the totient of any integer t is the product of totients of primes powers dividing t.

Now let      m=2n.

Since the only prime powers dividing m are 2 and n, φ(m)=(2−1)∗(n−1) = φ(m)=k, therefore φ(m)=φ(n)=k.

Journal ArticleThe Number of Solutions of φ (x) = m
Kevin Ford
Annals of Mathematics
Second Series, Vol. 150, No. 1 (Jul., 1999), pp. 283-311
Ford, Kevin. "The Number of Solutions of φ (x) = M." Annals of Mathematics, Second Series, 150, no. 1 (1999): 283-311. doi:10.2307/121103.

1.  definition, conjecture, postulate, theorem and undefined term :


2. Euler's Phi totient function:


3. Carmichael's number:


The Theory of Numbers, By R... by on Scribd


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