Science Forums 2

[Don Blazys]

To: CraigD. Thats good. Just exellent! You have more than redeemed yourself! I too apologize and take full responsibility for not being able to write it in LaTex. I work at a high school, and have now resolved to get one of the computer teachers to show me how to do it the first chance I get. As you could probably surmise, I'm quite inexperienced at working with computers, so I greatly appreciate your hard work and patience, and thank you for seeing it through rather than giving up like so many others have done.

Since we have now established that the first equation in post #12 is indeed an identity, I can promise that from here on out, that identity will make for a very interesting discussion, mainly because it is relatively unknown and not yet in any book, journal, magazine or encyclopedia. You see, I discovered it only about a decade ago, and it's consequences and ramifications have yet to be fully addressed by the math community. In fact, it's been only in the past few months that the math departments at two major universities and the editors of a major math journal have begun a serious study of this rather remarkable identity, so it is clear that we are all entering into "uncharted mathematical territory", where there is no "general consensus" on some very basic issues.

Personally, I find this all very exiting because this stuff is both "elementary", and "cutting edge", and just as the early experiments involving quantum mechanics required "intelligent interpretation", so does this relatively new identity!

For instance, since the first equation in post #12 is a proven identity, and since that identity is true for all values of the variables T={2,3,4...}, a={1,2,3...} and x={0,1,2...}, is the term on the right more suitable for representing "common factors" than the term on the left? I say yes, because "common factors" absolutely must be defined as being greater than unity in order for that concept to make any sense whatsoever, and the term on the right clearly precludes (prevents) T=1 when T is to be construed as a "common factor". What do you say?

Also, is it now proper to "cross out" the cancelled T's in the term on the left as you did in the second equation in post#12? I have a problem with that because once the T's are "crossed out", the term on the left appears as a^x, which can be construed as (1)a^x=(N/N)a^x where N does not equal T, and while the equation would remain true, it would no longer be an identity, and we would be either ambiguous (in the case of (1)a^x), or inconsistent (in the case of (N/N)a^x) as to the value of the cancelled factor or cancelled common factor. Again, what do you say?

There are other, even more serious issues, but let's discuss these two first. Maybe others will join us.

P.S. What kind of calculator do you have that is accurate to 18 decimal places, or is such accuracy available only in conjunction with a computer? Are they expensive?

Thanks again my mathematical friend!

Don.