is it possible that pi is a prime number?

is it possible that pi is a prime number?
or conversely, that the prime numbers are transcendental?

my thinking is this: it's impossible to define the prime numbers
in terms of anything other than themselves and 1, right? so if
there is no way to pin down pi numerically, can it too be a
prime number? in this case, the relationship is not between
the number itself and 1, as with the other primes, but rather
between the circumference and diameter of a circle.

not really know where i'm going with this ... it just seemed
like an interesting thought regarding number theory, about
which i know so little ...

-monz

monz wrote:

> is it possible that pi is a prime number?
> or conversely, that the prime numbers are transcendental?
>
> my thinking is this: it's impossible to define the prime numbers
> in terms of anything other than themselves and 1, right? so if
> there is no way to pin down pi numerically, can it too be a
> prime number? in this case, the relationship is not between
> the number itself and 1, as with the other primes, but rather
> between the circumference and diameter of a circle.
>
> not really know where i'm going with this ... it just seemed
> like an interesting thought regarding number theory, about
> which i know so little ...

I am not much of an expert on math, but I do know that pi and trig functions play into complex number theory. Take for instance the eighth root of 1. Actually there are eight possible solutions to 1^(1/8): 1, -1, i, -i, q+qi, q-qi, -q+qi and -q-qi, where q = 1/2^0.5 (the square root of one half). And you probably know what i is; it's the absolute value of the square root of -1, the base of imaginary numbers.

The eight solutions are all derived from dividing a circle into eight equal angles, all pi/4, and finding the sine and cosine of each multiple of pi/4.

I don't know if this has anything to do with what you're thinking, but yes, pi is indeed a fundamental number in physics, not just pure mathematics. And I have attempted to derive tunings from both pi and e (the natural logarithm base). I've already discovered the 87-tone pi-period e-generator scale, which is almost 53-tone equal temperament (the octave is stretched by about 2.5 cents). I've also suggested the natural logarithm function as a unit of measurement, calling it an "euler" (divided into "millieulers", each measuring 1.731234 cents). Other than being a completely unbiased form of scale measurement, neither octave-based or twelfth-based or anything, I don't see much other practicality in it, except as a novelty.

~Danny~

--- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:
>
> is it possible that pi is a prime number?
> or conversely, that the prime numbers are transcendental?

No, but there is a relativization in number theory of the idea of
"prime" to a particular algebraic number field. In the rational
numbers Q, 5 is a prime. However, in Q(phi), the Golden Ratio field, 5
is no longer a prime number, since sqrt(5) = 2 phi-1, and so 5
factors as (2 phi -1)^2. Nor is 11 a prime number, as 11 =
(3+phi)(4-phi). Instead of merely having 1 and -1 as what are called
"units", we have +-phi^n, for any integer n.

> my thinking is this: it's impossible to define the prime numbers
> in terms of anything other than themselves and 1, right? so if
> there is no way to pin down pi numerically, can it too be a
> prime number? in this case, the relationship is not between
> the number itself and 1, as with the other primes, but rather
> between the circumference and diameter of a circle.

Pi is a transcendental number, which means number theoretically in
some ways you can think of it as any arbitrary number "x". You can
write out algebraic expressions such as (sqrt(x)+1)/(x+1)^2 in terms
of x, and if they are distinct expressions algebraically, they will
still be distinct if you replace x with pi. And vice versa.