--- In

primenumbers@yahoogroups.com,

"David Broadhurst" <d.broadhurst@...> wrote:

> On Planet Zog the sloths are bipeds, all having the same

> number of toes, distributed between their two feet,

> with each foot having at least one toe. All possible

> sub-species occur. Each sloth has a composite number

> of toes on one foot, but not on the other.

Clearly each sloth has at least 11 toes,

since no smaller number will do.

Kermit sent me, off-line, the germ of the

proof that each sloth has *precisely* 11 toes.

Here is my own proof.

Let N be the number of toes of each sloth.

1) N is odd, else a sub-species of

sloth would have N/2 toes on each foot,

which is impossible, since no number

may be both composite and non-composite.

2) We know that N - 2 is composite,

since 2 is prime. Hence N > 9.

3) Since N > 9, we require that

N - 4, N - 6, N - 8 are all prime.

But one those 3 numbers must be divisible by 3.

The only prime divisible by 3 is 3.

Hence N - 8 = 3.

So N = 11 and we are done.

Comment: My anti-theological bias derives from

http://www.gutenberg.org/dirs/etext97/gltrv10.txt
"Which two mighty powers have, as I was going to tell you,

been engaged in a most obstinate war for six-and-thirty

moons past. It began upon the following occasion. It is

allowed on all hands, that the primitive way of breaking

eggs, before we eat them, was upon the larger end; but his

present majesty's grandfather, while he was a boy, going to

eat an egg, and breaking it according to the ancient

practice, happened to cut one of his fingers. Whereupon the

emperor his father published an edict, commanding all his

subjects, upon great penalties, to break the smaller end of

their eggs. The people so highly resented this law, that

our histories tell us, there have been six rebellions raised

on that account; wherein one emperor lost his life, and

another his crown. These civil commotions were constantly

fomented by the monarchs of Blefuscu; and when they were

quelled, the exiles always fled for refuge to that empire.

It is computed that eleven thousand persons have at several

times suffered death, rather than submit to break their eggs

at the smaller end. Many hundred large volumes have been

published upon this controversy: but the books of the

Big-endians have been long forbidden, and the whole party

rendered incapable by law of holding employments."

Fortunately, debate on the non-primality of units

has not, so far as I am aware, led to any fatality.

David (per proxy J. Swift)