I was reading my copy of the new American Mathematical Monthly. David Cox and
Jerry Shurman wrote an interesting article on "Geometry and Number Theory on
Clovers". It generalizes both Gauss' proof that you can subdivide a circle into
n parts using ruler and compass iff n is a power of two times a product of
distinct Fermat primes and Pierpont's 1895 result that you can divide a circle
into n parts using origami if and only if n is a product of a power of two times
a power of three times a distinct product of primes of the form 2^n*3^m+1.
(These primes are now called Pierpont primes.)

But here is the funny part, on page 3 of this fine article they write:

Gleason suggests that there may be infinitely many Pierpont
primes, although only finitely many have been found so far.

"Only finitely many found," grin, that is true for primes in general! so for
every defined type of primes! Then they continue the unintended humor with

According to Sequence A005109 of Sloane's... the only known
Pierpont primes form the set:

Here they list the 40 such primes below 1,000,000 as if somehow this could
possibly be all
known Pierpont primes! Sometimes excellent mathematicians do not pause to think
about the computational questions.

The Wikipedia article http://en.wikipedia.org/wiki/Pierpont_prime does a little
better:

As of 2005, the largest known Pierpont prime is 33853318889473 .

and then misstates Pierponts theorem. But of course we expect much less from
Wikipedia.

Both statement are obviously silly. It is trivial to find bigger Pierpont
primes. Here are a couple quick (to program) counts I did with pari's isprime:

------ --------------
Pierpont primes
N below N
------ --------------
10^1 2
10^2 8
10^4 23
10^8 56
10^16 123
10^32 248
10^64 503
10^128 1018
10^256 2073
10^512 4225
------ --------------

I used doubling exponents to show clearly the number Pierpont primes (appear to)
grow with the simple pattern you'd expect. I used ispseudoprime to calculate
the value to 10^1024, but turned the machine off without recording the result.
I guess more than 40 are known!


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