Some "ABC good" intervals

I found a web page, http://www.minet.uni-jena.de/~aros/abc.html, with
a list of all 148 known "good" ABCs according to the definition you
will find there of "good" (there are others.) Cutting things off at
the 53-limit and keeping A small, I found the following, with names
if Manuel gives one on his web page:

(5) 4375/4374 = 5^4*7 / 2*3^7 "ragisma" (ragisma??)

(17) 32805/32768 = 3^8*5 / 2^15 = shisma

(19) 301327048/301327047 = 2^3*11*23*53^3 / 3^16*7

(30) 2401/2400 = 7^4 / 2^5*3*5^2 = "breedsma", presumably after our
very own Graham Breed. Congratulations, Graham!

(38) 63927525376/63927525375 = 2^13*11^4*13 *41 / 3^3*5^3*7^7*23

(47) 512001/512000 = 3^5*7^2*43 / 2^12 * 5^3 (How's 2^12*5^3 /
3^5*7^2 for an approximation to 43 in the 7-limit? It's all of 1/296
cents flat.)

(82) 128/125 = 2^7/5^3 = enharmonic diesis

(131) 282192773751/282192773120 = 3^3*7^10*37 / 2^26*5*29^2

Numero Uno on the list? 6436343/6436341 = 23^5 / 3^10*109; remember
that when you work in the 109-limit. Of course, many of these are not
best thought of in terms of the p-limit, but in the set of numbers
generated by their prime factors, and toss in 2.

--- In tuning-math@y..., genewardsmith@j... wrote:
> I found a web page, http://www.minet.uni-jena.de/~aros/abc.html,
with
> a list of all 148 known "good" ABCs according to the definition you
> will find there of "good" (there are others.)

I don't see a definition there of "good", and most of these are not
superparticular ratios at all . . . so what, really, is the ABC
conjecture?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> I don't see a definition there of "good", and most of these are not
> superparticular ratios at all . . . so what, really, is the ABC
> conjecture?

If we have three relatively prime positive integers A, B, C such that
A + B = C, and if we define a "radical" function rad(N) as the
product of the primes dividing N, we can look at C/rad(ABC); it turns
out that this can be arbitarily large. The ABC conjecture says that
C/rad(ABC)^e, for any e>1, cannot become arbitrarily large. It's a
conjecture in elementary number theory with a large number of very
powerful and not always elementary consequences, and since it is the
latest fad I thought of it when I saw that the list of m/n in the
p-limit, with |m-n|<d, should be finite. It seems it is also an easy
consequence of Baker's theorem, and hence is true--a useful thing to
know. Any ABC triple such that ln(C)/ln(rad(ABC)) > 1.4 is rather
arbitarily termed "good"; they are pretty rare and some of them turn
up in music theory, it seems. For superparticular ratios this measure
becomes ln(C)/ln(rad(BC)), and looking at the "goodness" of our
favorite commas might be interesting, I suppose.