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

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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.

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

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2. Euler's Phi totient function:

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3. Carmichael's number:

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

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