Problems & Puzzles:
A bound to the
largest prime factor of certain Carmichael numbers
A number n is said to be a "Carmichael
* x^(n-1)=1 mod n, for all x such that:
* gcd(x, n)=1
* n= an odd composite number having 3 or more prime factors.
It can be shown that the above definition is equivalent to say that for
number n, for every prime p that divides n then p-1 divides n-1 also.
From now on, let's discuss only about the C3
odd numbers composed of 3 prime factors such that C3
= n = p*q*r,
A few examples are:
At the p.44, of "Recreations in the Theory of
Numbers" by Albert H. Beiler we can read:
"if... p is assigned value then there are only a
finite number of values for the other two primes [q & r];
they can not be increased indefinitely"
The same idea is said by Eric Weisstein:
"For Carmichael numbers with exactly three prime
factors, once one
of the primes
has been specified, there are only a finite number of Carmichael numbers
which can be constructed." (See his article http://mathworld.wolfram.com/CarmichaelNumber.html)
Then there are only finite values n, if any, for that p value.
But then also it's true that for each p, r has some upper limit.
1) Can you explain the Bailer
claims regarding the finitude of the
numbers for a
How large is r against p?
3) A simple inspection of the first C3
numbers (*) shows that r < p3
(let's take this as a starting conjecture to improve). The same data shows that
very soon this bound is too
wide... Exist a better (tighter) and simpler bound?
(*) See the car3-18 file from the Richard Pinch's site: ftp://ftp.dpmms.cam.ac.uk/pub/Carmichael/
See other problem related to Carmichael numbers in this
site at Problem 24.
The Saturday 6 of May both Richard Pinch and Chris
Nash solved the question of ths puzzle-conjecture, basically following
the same reasoning and arriving to the same "better bound". The
argument, not only the conclusion is interesting. Here are complete:
Richard Pinch (10:00 am) wrote:
"Let N = pqr be a Carmichael number with three prime factors
p < q < r and assume p is given. We have N == 1 mod (p-1), mod
(q-1), mod (r-1). Hence qr == 1 mod (p-1), pr == 1 mod (q-1) and pq == 1
mod (r-1). Say pr-1 = a(q-1) and pq-1 = b(r-1).
Since q < r, we have q < r-1 and so p(r-1) > p(r-1)-1
> pq-1 = b(r-1). Hence b < p. Similarly, a(q-1) = pr-1 > p(q-1)
so p > a. In fact we see that b=1 is impossible, otherwise we would
have pq-1 = r-1, that is, pq=r, but r is prime. Put D = ab - p^2. After
a little algebra, we find that q-1 = (p-1)(p+b) / D and r-1 =
(p-1)(p+a)/D, which shows that D > 0 and that D = (p-1)(p+b)/(q-1)
< p+b < 2p. So there are only a fixed number of possible values of
D. For each value of D, there are only finitely many values of a and b,
since ab = D+p^2. Hence the are only finitely many pairs (a,b) and hence
(q,r) for given p.
Further, we can bound r as follows. a = (D+p^2)/b and a > p. So
(p+a)/D < 2a/D < 2(D+p^2)/bD < 2/b + 2p^2/bD <= 1 + p^2. So
r-1 < (p^2+1)(p-1) = p^3-p^2+p-1 and so r < p^3-p^2+p < p^3. So
(p+a)/D < p/D + (D+p^2)/bD = p/D + 1/b + p^2/bD <= (p^2+2p+1)/2 as
b >= 2. So r-1 <= (p-1)(p+1)^2/2 = (p^3+p^2-p-1)/2..."
Chris Nash (1:36 pm) wrote:
"If pqr is a Carmichael number, then the following
must all be integers. In fact, they must all be greater than 1.
I_1 is "a little greater" than rp/q. I_2 is "a little
greater" than pq/r. I_1*I_2 must also be an integer. I_1*I_2 is
"a little greater" than (rp/q)*(pq/r)=p^2. So all you need to
prove is that I_1*I_2 is less than p^2+1. (So then we have an integer
I_1*I_2 that is between p^2 and p^2+1, which is impossible).
Let p be a known prime, and choose a prime r. We need to find
conditions on r to prove a solution does not exist. We proceed by
solving for p, q, and r for every possible value of I_2 (all integers
greater than one).
Pick I_2, and calculate k such that r=(k.p^3+p-1)/I_2 + 1 Then q=k.p^2+1
by equation (3). By equation (2) I_1 = (rp-1)/(q-1) = (p2+1/k-1/kp)/I_2
+ (p-1)/(kp^2) and must be an integer.
Thus I_1.I_2 = p^2 + 1/k - 1/kp + I_2(p-1)/(kp^2) must also be an
integer. Since p^2 is an integer, no solution exists if 0< 1/k - 1/kp
+ I_2(p-1)/(kp^2) < 1
And so no solution exists if k > 1 - 1/p + I_2(p-1)/(p^2) and
hence for every possible choice of I_2, there is a limiting value of r,
beyond which no solutions exist. Substitute this value of k into the
formula for r.
Thus for each I_2, no solution exists for r > (p^3 - p^2 +
I_2.p.(p-1) + p - 1)/I_2 + 1 = (p^2 - p + 1) + (p^3 -p^2 + p - 1)/I_2.
This is a decreasing function of I_2. So a tight bound for r is
found by putting I_2=2... (then) r<=(p^2-p+1) + (p^3 - p^2 + p
- 1)/2 = (p^3 + p^2 -p + 1)/2"
Both arguments reach to the
same bound: r<=(p^3+p^2-p+1)/2
But they have obtained some additional
things than the originally asked.
When I asked to Richard Pinch
if a bound for q against p can be established he replied (2:43 pm) "Yes:
I claim q < 2p^2. Using the same notation, one has q-1 = (p-1)(p+b)/D
and b < p. So q-1 < (p-1).2p ...."
This means that q<2p^2-2p+1
By his side Chris Nash answered
(7/5/2000, 10:12 am) to the same question: "... q<=
2p^2 - 3p +2 and make the comment that this bound cannot be
attained unless p=3 ". (Does anybody wants the complete
In this case Richard
& Chris have reached to two slightly different results being
the Chris's bound a little bit more tight than the Richard's
Regarding the limit of bound for r Chris
Nash wrote (1:36 pm):
"...The (limit of the)
bound can only be attained if this is prime, and if I_2=2, i.e. if
q=p^2+p-1 is also prime and satisfies the conditions. It is not a
simpler bound, but it is definitely a tight bound. For instance, if p=3,
the tight bound is (27+9-3+1)/2=17. And indeed, 3 *11*17 is a Carmichael
number, so this bound is attained. p=5, the bound is (125+25-5+1)/2,
which is 73. 5*29*73 is indeed a Carmichael number. The bound is
attained yet again... To attain the
bound, these three numbers need to be the prime factors: p, p^2+p-1 and
(p^3+p^2-p+1)/2... Of course it is unlikely provable that an infinity of
such prime triplets exist. Perhaps you would be interested in searching
for the largest Carmichael number of this form?"
Later (7/05/2000, 4:19 am), Richard
should have added that the proof of the finiteness of qr given p is not
original to myself: I think it in Carmichael's paper of 1912,
referred to in Beiler's book. I did however give a version of it,
similar to the one I sent you, in my paper Carmichael numbers up to
10^15, Math. Comp. 61:203 (Jul 1993) 381--391, which can be found on
the net via URL http://www.chalcedon.demon.co.uk/publish.html#37
So we have two new questions:
a) are there infinite triplets of extreme
primes p, p^2+p-1 and (p^3+p^2-p+1)/2
b) find larger and larger of such extreme triplets.
Just as a starting point I tackled the
task of finding a kind of moderate large of such extreme
triplets, hoping that readers may soon and happily beat it:
p = 7352265*2^400+1 (is prime)
q = p^2+p-1 (is prime)
r = (p^3+P^2-p+1)/2 (is prime)
Then we have here (Sunday morning) a
C3 = p.q.r = an extreme Carmichael C3
number of 764 digits