* * See also
* conject * conjecturalVerb(conjectur)
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.
Suppose n is prime and n−1= k
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
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:
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