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.