Carmichael's Theorem/Mistake

Source Work

1986: David Wells: Curious and Interesting Numbers:

The Dictionary

$144$

1997: David Wells: Curious and Interesting Numbers (2nd ed.):

The Dictionary

$144$

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:

Let $n \in \Z$ such that $n > 12$.

Then the $n$th Fibonacci number $F_n$ has at least one prime factor which does not divide any smaller Fibonacci number.

The exceptions for $n \le 12$ are:

$F_1 = 1, F_2 = 1$: neither have any prime factors

$F_6 = 8$ whose only prime factor is $2$ which is $F_3$

$F_{12} = 144$ whose only prime factors are $2$ (which is $F_3$) and $3$ (which is $F_4$).

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $144$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $144$