Primes to Non-Integer Powers

This is a "prime" website for "prime" info:

At: http://www.math.umd.edu/~krc/numbers/fta.html

it says:

<<the Fundamental Theorem of Arithmetic.>>

Note that this title above is *NOT* meant to be interpreted as saying: "because we are using the term 'fundamental" as well as the term 'arithmetic" in the title of this axiom, this axium can be accurately applied to - *all* operations arithmetic...

Let's ignore the (here) irrelevant detail that we are often dealing with a ratio of two numbers, and focus on the factorization of a single (numerator or denominator) number only.

This axiom, and any assumptions thus arising out of this axiom, are applicable *ONLY* in the case of prime number factors to an integer (non-rational power).

The fact that <<the Fundamental Theorem of Arithmetic.>> uses the word "fundamental" only implies its "importance" (in the human, more than numerical sense)!

Its properties are true *only under certain conditions* (as opposed to it's properties being applicable in all cases of the factorizations of numbers - as with the fractional exponents!! :)

Prime Number:
http://www.math.umd.edu/~krc/numbers/prime.html

<<Definition: An integer p is called a prime number if the only positive integers that divide p are 1 and p itself. Integers that are not prime are called composite.>>

Note that prime numbers are *integers* by definition. If we combine three separate factors of 3 x 3 x 3 into a notational convenience of 3^(3), we are not changing the fact that (the factors being integers), the product will *always* be an *integer*.

Such is not the case with (ie) 3^(3/2) = 5.19615242271, which may turn out to have either:

(1) Repeating digits [at which point, by the FTA, it must be factorable into an (integer valued) rational number - by first multiplying it's value by 10^(E) where E is the power of 10 nesccessary to restore it to an integer, then factoring (by the FTA) to prime factors to *integer* powers] ; OR

(2) Non-repeating digits, meaning it is "transcendental" [such as 2^(1/2), phi, pi, e].
_________________

From the page: http://www.math.umd.edu/~krc/numbers/fta.html

<<the Fundamental Theorem of Arithmetic.>>
<<Theorem: Every integer N > 1 can be written uniquely as a product of finitely many prime numbers.>>

The term "uniquely" is *NOT* intended here to be interpreted in the "human sense" as "novel", etc. It means ONLY in THAT CASE! The axiom does not address, nor guarantee "uniqueness" (meaning one and ONLY ONE solution) in ANY OTHER case other than the *integer* case which it declares!

If we meet "5.19615242271" on the road, we don't exclaim, "oh hi, you're '3^(3/2)', aren't you?". Such would be an incredible coincidence, indeed. By the same characteristics of nature, I was not able to easily construct rational exponents of prime numbers which (in 2 separate and different examples) would both equal an integer taken to an *integer* power.

If you setup your Excel "solver" (as an "iterative" approach), or use the linear algebra (which you and lots of folks are more adept than I at!) to solve for (ie) 7^(x) * 5^(y) = 3^(1) * 2^(-1)
[ in canonical form: (7^(x) * 5^(y)) - (3^(1) * 2^(1)) = 0 ], you will discover an endless results array of possible combinations of 7^(x) * 5^(y) which will (within machine's numerical precision) equal 3/2. Just set 'x' equal to N values between 0 and infinity, and for *each* individual (real) value of 'x' you will find a corresponding (unique, real) value of 'y'. An "infinite universe" of (hard to find by "peckin-around"), but nevertheless *real* potential pairs of 'x' and 'y' which will *all* come within 0.00000000001 CENTS of a "perfect" 3/2 !!!

If numerical perfection is required, you *have to* stick with "integer" (only) calculations. That is an "even more" fundamental "rule" of (the numerical precision of) arithmetic operations.

And that's exactly what makes the "Fundamental Theorem of Arithmetic" so powerful and unique! Unfortunately, it's "integers-only" (or the identical *integer* case of multiples of, or powers of the (prime class) *integers*.

Note (also), in relation to the number One *not* being considered to be an *actual* "prime number", that: http://www.math.umd.edu/~krc/numbers/fta.html
states that:

<<The statement of the Fundamental Theorem of Arithmetic only talks about integers greater than one. The main reason for this restriction is that the integer 1 is not a prime number. Oh, it meets the criteria presented in the usual careless definition (only divisible by 1 and itself), but careful mathematicians always explicitly exclude 1 from the list of primes.>>

<<Now, having excluded 1 from the list of primes, you certainly can't write 1 as a product of other primes (all of which are bigger than 1). Well, you could if you allowed a product of zero primes, but that seems too much like cheating.>>

<<The main point of the Fundamental Theorem of Mathematics is the uniqueness of factorizations. It is relatively easy to show that some kind of factorization into primes exists; it takes considerably more care to show that there is only one way to factor an integer.>>

The FTA (which they also slip into calling the FTM, above) is a blessing (in that it declares the special *integer* case to work for factorizations), but (from "solver" example, above), does not imply *uniqueness* in the case of prime factors to non-integer exponents.

I believe that is why Graham, when asked about lattices in relation to his mean-tone tempering algorithms has (in every instance I have seen on these lists) stated that these quantities are "not applicable" (or something to that effect) to "lattices"?

Seems to me that these structures are premised upon the concept of the "uniqueness" of a any "tile" or "node" as described by *one* (and *only* one) combination of primes taken to given powers (which we find only in the *integer* case).

J Gill :)

>.
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>Please, give concrete actual examples.
>
>
>-monz
>
>
>---- original message --------
>
>From: unidala <JGill99@imajis.com>
>To: <joemonz@yahoo.com>
>Sent: Friday, December 28, 2001 11:39 PM
>Subject: Fwd: Re: super superparticulars
>
>OFFLIST
>
>Joe,
>
>Got your email. Thanks for your thoughts.
>
>With regards to the response below. Don't know if you have communicated
>further with folks about it, but the deal (I believe) with rational *powers*
>of primes as the factors (which multiplied) yield "any number" is - that
>*multiple combinations* [of the *same* primes with different values of
>rational exponents, or (possibly) of *different* primes with different
>values of rational exponents] MAY ALSO be exactly EQUAL to the SAME NUMBER
>as a result of their being so combined. Therefore, one *cannot* (in the
>circumstance of non-integer powers of the exponents) state that there is ONE
>and ONLY ONE *solution* for factoring the original integer (or, in our case
>the ratio of integers) numerical value.
> >
> > While one might say, "great, that means even more possibilities for
>factoring the (in our case, rational) number!", it *is* significant, in that
>(in the the circumstance of non-integer powers of the exponents) NUMEROUS
>DIFFERENT combinations (and an indeterminate amount of them, to boot) WILL
>YIELD THE SAME RATIONAL NUMBER!
> >
> > Therefore (if you *do* use such non-integer powers of the exponents), THEN
>YOU CANNOT *UNIQUELY* SPECIFY A GIVEN RATIONAL NUMBER. So, whatever
>SPECIFICITY your (various primes to positive and negative non-integer
>powers) *APPEAR* to imply (in denoting a given rational number such as a
>scale pitch in a periodicity block) IS FALSE (because there are at least
>one, if not many more, different, thus *NON-UNIQUE*, and *other* ways the
>same rational number can be denoted.
> >
> > The result being - the pitch ratio you are denoting (when using
>non-integer exponents of the prime factors) can be denoted by one or more
>other combinations of *other exponents* and/or *other primes*. Which, then
>do you decide upon - not to mention that there is no "closed-form"
>arithmetic solution (other than "brute-force" numerical iteration which you
>could *never* state had exhausted all possible solutions, because it cannot
>be *known* how many solutions exist).
> >
> > Clearly, you could simply find the combination of a group of any primes
>(pick any primes you like!) which happen to have the particular non-integer
>exponents which (when combined) yield the pitch ratio which you would like
>to equal (or approximate).
> >
> > Do you see the problem, though. For example, Joe Monzo might like
>(hypothetically) 2^(a/b)* 3^(c/d) to equal Ratio (R). But Jeremy prefers
>17^(e/f)* 31^(g/h), which *also* is EXACTLY equal to the *same* pitch ratio.
>Note how neither of us would be wrong. Conversely, NEITHER OF US WOULD BE
>"RIGHT", EITHER (because there is no longer a *UNIQUE* (one and only one)
>solution to the problem!!!
> >
> > Now this is fine UNLESS you want to "tile" it into one of these darn
>"periodicity blocks" (where a "unique" solution for each individual pitch
>ratio is implied and required).
> >
> > One solution - ignore "periodicity blocks" (despite what people may
>think), and enjoy finding "non-unique" solutions [with the least amount of
>low valued numbers (which no longer even have to be prime!) taken to the
>powers of your liking which *happen* to equal the pitch ratio. But, you
>would have to proceed by "iteration" (hunt and peck) somehow (since there is
>no way an arithmetic algorithm can determine all solutions, or even
>determine how possible solutions exist), and (even if you found the smallest
>values of integers (prime or otherwise) which (when taken to certain
>rational exponents) yielded your desired pitch-ratio, that would represent
>only *ONE* out of an *INDETERMINATE* number of other possible solutions. As
>long as you are comfortable with those circumstances, viva la difference!
>:)
>
>
>
>_________________________________________________________
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