```From:           Bill Daly <bill.daly@tradition-ny.com>
To:             sci.math
Subject:        Re: Perfect numbers and Carmichael numbers - a hidden relation
Date:           Thu, 25 Jan 2001 12:23:42 -0800
```

```In article <94no7t\$37v\$1@nnrp1.deja.com>,
tim_robinson@my-deja.com wrote:
> ...
> This is an important result.   You should submit it for publication.

Actually, it is a special case of a much more general result. Suppose
that N has a set of distinct divisors d[1]..d[k] whose sum is divisible
by N. Suppose that for some x the numbers p[i] = x*N*d[i]+1 are all
prime. Let A be the product of the p[i]. Then A is a Carmichael number.
It is easy to show that A-1 is divisible by x*N^2, and that x*N^2 is
divisible by x*N*d[i] = p[i]-1 for all i, thus the necessary conditions
are satisfied.

For example, 20 = 1+4+5+10, thus if 20x+1, 80x+1, 100x+1 and 200x+1 are
all prime, then their product is a Carmichael number. This is the case
for x=333 and x=741.

It is relatively easy to find such partitions of N. If rN =
d[1]+...+d[k], then dividing by N we get r = 1/e[1]+...+1/e[k], where
e[i] = N/d[i]. Thus, every Egyptian fraction representation of the
integer r will lead to a partition of the desired type. The example
above is derived from 1 = 1/2 + 1/4 + 1/5 + 1/20.

Note that if the set of divisors d[1]..d[k] does not include N itself,
then we can always append d[k+1] = N to the set. Thus, for example, not
only is (6x+1)(12x+1)(18x+1) a Carmichael number whenever the three
factors are prime, but also (6x+1)(12x+1)(18x+1)(36x+1) is a Carmichael
number if in addition 36x+1 is prime. For example, both 7*13*19 and
7*13*19*37 are Carmichael numbers.

An interesting possibility is the following. There are numbers N for
which there are two (or more) distinct sets of divisors whose sums are
divisible by N. (By distinct, I mean having no elements in common.) For
example, 120 = 20+40+60 = 1+2+3+4+5+6+8+10+12+15+24+30. If we can find
an x such that the corresponding p[i] are all prime, then we will have
found a Carmichael number which is the product of two smaller Carmichael
numbers. Another possibility (with fewer divisors) is 360 = 60+120+180 =
5+12+36+40+45+60+72+90.

Regards,

Bill

Sent via Deja.com
http://www.deja.com/
```

```From:           Bill Daly <bill.daly@tradition-ny.com>
To:             sci.math
Subject:        Re: Perfect numbers and Carmichael numbers - a hidden relation
Date:           Thu, 25 Jan 2001 18:02:37 -0800
```

```In article <94pvi9\$7k\$1@nnrp1.deja.com>,