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I'm having some trouble while trying to prove the well known fact that a Carmichael number has at least 3 prime factors.

Basically, how I see it, I have 2 options:

  1. building a number b that will satisfy $b^{n-1}\not\equiv 1(n)$ thus creating a fermat witness

  2. trying to contradict the condition of: $p|n \iff\ p-1|n-1$

Maybe I'm missing something very simple but I can't seem to get this done.

Addition: we are talking about the case where $n=pq$, the case $n=p$ is trivial.

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3 Answers 3

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Assume that $n=pq$, with $p<q$ two distinct primes, is a Carmichael number. Then we have $$ q\equiv1\pmod{q-1}\implies n=pq\equiv p\mod{q-1}\implies n-1\equiv p-1\pmod{q-1}. $$ Here $0<p-1<q-1$, so $n-1$ is not divisible by $q-1$. A contradiction.

The same argument can be extended to prove that any prime factor of a Carmichael number $n$ is less than $\sqrt n$. Namely, as above we get for all primes $p\mid n$ $$ n-1\equiv p\frac{n}{p}-1\equiv \frac{n}{p}-1\pmod{p-1}. $$ Therefore $n/p-1$ must be a (positive) multiple of $p-1$. Thus $$ \frac{n}{p}-1\ge p-1\implies\frac{n}{p}\ge p\implies n\ge p^2. $$

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In a carmicheal number, you need at least three prime factors. These primes might be written in the form $p_x = a_x n + 1$ where $n$ is the common divisor of $p_x-1$.

If there were just two prime factors, this would expand to $a_1 a_2 n^2 + (a_1 + a_2)n + 1$. This number is required to be a multiple of both $a_1$ and $a_2$, but it can not be, when $a_1$ and $a_2$ are un-equal, and it won't work when it is (because the maximum period of a square of $p$ is divisible by $p$).

In order to make the term on $n$ be a multiple of $a_j$ over all $j$, you need at least three separate primes, so that any two can be complementry relative to the third. For example, in a base where $1,1$, $2,1$, and $3,1$ are all prime, then the product of pairs are not divisible by $p-1$, but in the product of all 3, you have $6,11,6,1$, which is divisible by $1,0$, $2,0$ and $3,0$, since the base would be a multiple of $6$ anyway.

Carmichael Numbers and Primes

These are involved in the process test of large number for primeness.

The simple test of dividing $n$ by every $p$ up to the square root, has a cost in the order of the square root of $n$. For $n$ of the order of $1,000,000$, this can cost, eg £1000. There are other tests that can reduce the cost to eg the cost per digit: ie £6.

The oldest of these is that for any number $x$, $x^p \mod p = x$ is true whenever p is prime, and if it is not true, then $p$ is definitely composite. You can see the results by looking at the remainders of $2^p$, when divided by $p$.

     3   4   5   6   7   8   9   10   11    12    13    n
     8  16  32  64 128 256 512 1024 2048  4096  8192   2^n
     2   0   2   4   2   0   8    4    2     4     2   remainder
     y   n   y   n   y   n   n    n    y     n     y   prime?

Raising 2 to large powers to find the remainder when divided by some $n$, can be made much easier if multiples of $n$ are cast out as one goes. This brings the calculations well into range of hand or desktop calculator.

For example, for 341, one might note that $2^{10} \mod 341 = 1$. One then calculates $1^{34}*2 \mod 341$ to get $2$. Although this is a marker of a prime, it is in fact composite. There is a very small proportion of composite numbers that sneak through the test. These are called pseudo-primes to $2$.

Most of the pseudo-primes to $2$ are not pseudo-primes to $3$ or some other number. So doing the test for two numbers, one will filter out most of the pseudo-primes. There is however a class of number that are pseudo-prime to all bases: the Carmichael numbers. These are rarer than pseudoprimes, but still one needs to be aware of these.

On the other hand, the number $5461 = 43 \cdot 127$ divides all $(2^a\;5^b)^{42n}-1$, this includes $5460$ itself, so doing a test over two different bases do not exactly remove all pseudoprimes. $5461$ is a pseudoprime to all bases $2^a\;5^b$.

The test for primeness supposes that if $p$ is prime, then some $b^{p-1}=1 \mod p$, for all $b$ co-prime to $p$. In a pseudo-prime, both $p$ and $q$, divide some $2^m-1$, while $m$ divides both $p-1$ and $q-1$. In the case of $341$ above, $m=10$, which divides both $11-1$ and $31-1$. The reason it usually fails, is that selecting a different base $b$, will mean that the period (the smallest $x$ that satisfies $p \mid b^x-1$), does not divide the other prime's $p-1$.

A Carmichael number is a pseudo-prime to every base. For a composite number, the period $\lambda = \operatorname{lcm}(p-1, q-1, \cdots...)$, where the composite number is $pq..$. The lcm can not divide both $pq-p-q$ since it would require $p-1$ to divide $q$, which does not happen. When three names are involved, one gets $\operatorname{lcm}(p-1, q-1, r-1)$, and it is then possible for any two to complement the third.

For example, $1729$ is a carmichael number. It has three factors: $7, \; 13, \; 19$. When one finds $\operatorname{lcm}(7-1, 13-1, 19-1)=36$, one sees that $36 \mid 1729-1$, and so one can't tell this by this test if it were prime or not. Of course, there are quite large numbers, like $601 \cdot 1201 \cdot 1801$, whose period is $3600$, divides the product of these numbers, less 1.

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    $\begingroup$ I find the term Carmichael primes to be very confusing, after all, they are not primes. Maybe, you should replace this by Carmichael numbers $\endgroup$
    – Tomas
    Jul 1, 2013 at 8:32
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(Personally, I find the following proof very unsatisfying, because it provides me no insight: it seems to all boil down to rather tedious numerical nitty-gritty. Unfortunately, this seems to be par for the course in number theory, at least in my limited experience.)

Suppose $n = pq$, where $p$ and $q$ are prime, $p < q$, and $(q - 1)|(n - 1)$. We show that these conditions lead to a contradiction.

The last condition is equivalent to saying that there exists an integer $k$ such that

$$ n - 1 = pq - 1 = k(q - 1)\, \tag{1} $$

Since $q > p > 1$, it follows that $n = pq > q$, and therefore $n - 1 > q - 1$.

This, together with $(1)$, implies that $k > 1$.

Now, rearranging the second equality in $(1)$ we get

$$ k = q(k - p) + 1 \,. $$

Since we already established that $k > 1$, the last equality implies that

$$k - p \geq 1\,.\tag{2}$$

Now, once more rearranging the second equality in $(1)$, to

$$ 0 = k(q - 1) - pq + 1\, $$

and adding $p$ to both sides, we arrive at a contradiction:

$$ p = k(q - 1) - pq + p + 1 = (k - p)(q - 1) + 1 > (k-p)(p-1) + 1 \geq (p-1) + 1 = p \,. $$

Note that the last inequality above follows from $(2)$.

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