Carmichael's Theorem/Mistake

Source Work

 * The Dictionary
 * $144$
 * $144$


 * The Dictionary
 * $144$
 * $144$

Mistake

 * A divisor of a Fibonacci number is called proper if it does not divide any smaller Fibonacci number. The only Fibonacci numbers that do not possess a proper divisor are $1$, $8$ and $144$.

The following points about the above are questionable.


 * $(1): \quad$ The concept of a proper divisor of a Fibonacci number defined in this way has not been corroborated by means of an internet search. A proper divisor of an integer has a different definition.


 * $(2): \quad$ Having defined a proper divisor of a Fibonacci number as above, it needs to be pointed out that $4$ is a proper divisor of $F_6 = 8$, as no Fibonacci number smaller than $8$ has $4$ as a divisor.  Similarly, $F_{12} = 144$ has the proper divisor $72$, which, similarly, is not the divisor of any Fibonacci number smaller than $144$.


 * $(3): \quad$ In its definition, a proper divisor of a Fibonacci number is not directly specified as having to be a proper divisor, in the conventional sense that a proper divisor of $n$ is a divisor of $n$ which is not $1$ and is not $n$.


 * $(4): \quad$ The only divisor of $1$ is $1$. But $1$ vacuously is a divisor of no smaller Fibonacci numbers, and so, again by the above definition, is a proper divisor of $1$.

The conventional presentation of this result is Carmichael's Theorem, to discuss the primitive prime factors of a Fibonacci Number: