It states that for integers a+b=c, the ratio of sqp(abc)^r/c always has some minimum value greater than zero for any value of r greater than 1. For example, if a=3 and b=125, so that c=128, then sqp(abc)=30 and sqp(abc)^2/c = 900/128. In this case, in which r=2, sqp(abc)^r/c is nearly always greater than 1, and always greater than zero.
Obviously I'm reading this wrong -- because as stated (and assuming that a, b, and c are positive integers) this seems trivially true -- sqp(abc) cannot be zero, r cannot be negative, and c is finite, so therefore sqp(abc)^r/c is greater than zero, QED.
Does Nature mean that the quantity does not approach zero as r tends to infinity (or some such)? Their example sure doesn't seem to indicate such.
Yes, sqp(abc)^r/c is greater than 0. But what we are saying here is that it is _bounded_.
The formulation in the article is a little confusing/non-intuitive. A perhaps more 'layman' explanation is that for every r > 1, there are only finitely many coprime tuples (a,b,c) such that a+b = c and c > sqp(abc)^r
In other words, in "most" cases, sqp(abc) is greater than c!
This is equivalent to the definition in the article. Why? Well, given a fixed r, there are finitely many tuples that satisfy c > sqp(abc)^r. Since there are finitely many, we can pick a constant K > 0 st c < K*sqp(abc)^r for ALL tuples (a,b,c). Thus:
Thanks, after reading that through a couple of times, I think I understand what it means.
Looking back at the article, this is the critical bit:
> the ratio of sqp(abc)^r/c always has some minimum value greater than zero for any value of r greater than 1
That looks like it's just saying sqp(abc)^r/c > 0. But from what you're saying, the bit about the minimum value (1/K) is important. For each value of r, it should be possible to define a minimum value greater than 0 which sqp(abc)^r/c can never be smaller than. Right?
Are those minima interesting in themselves, or just as part of the whole conjecture? Is there a method for finding the minimum for a given value of r?
"It almost looks as if the method doesn't (as presently written) produce explicit constants. Hopefully as the ideas become more widely understood, effective constants will be able to be extracted."
Comments
It states that for integers a+b=c, the ratio of sqp(abc)^r/c always has some minimum value greater than zero for any value of r greater than 1. For example, if a=3 and b=125, so that c=128, then sqp(abc)=30 and sqp(abc)^2/c = 900/128. In this case, in which r=2, sqp(abc)^r/c is nearly always greater than 1, and always greater than zero.
Obviously I'm reading this wrong -- because as stated (and assuming that a, b, and c are positive integers) this seems trivially true -- sqp(abc) cannot be zero, r cannot be negative, and c is finite, so therefore sqp(abc)^r/c is greater than zero, QED.
Does Nature mean that the quantity does not approach zero as r tends to infinity (or some such)? Their example sure doesn't seem to indicate such.
Yes, sqp(abc)^r/c is greater than 0. But what we are saying here is that it is _bounded_.
The formulation in the article is a little confusing/non-intuitive. A perhaps more 'layman' explanation is that for every r > 1, there are only finitely many coprime tuples (a,b,c) such that a+b = c and c > sqp(abc)^r In other words, in "most" cases, sqp(abc) is greater than c!
This is equivalent to the definition in the article. Why? Well, given a fixed r, there are finitely many tuples that satisfy c > sqp(abc)^r. Since there are finitely many, we can pick a constant K > 0 st c < K*sqp(abc)^r for ALL tuples (a,b,c). Thus:
sqp(abc)^r/c > 1/K > 0, as the article mentions.
Thanks, after reading that through a couple of times, I think I understand what it means.
Looking back at the article, this is the critical bit:
> the ratio of sqp(abc)^r/c always has some minimum value greater than zero for any value of r greater than 1
That looks like it's just saying sqp(abc)^r/c > 0. But from what you're saying, the bit about the minimum value (1/K) is important. For each value of r, it should be possible to define a minimum value greater than 0 which sqp(abc)^r/c can never be smaller than. Right?
Are those minima interesting in themselves, or just as part of the whole conjecture? Is there a method for finding the minimum for a given value of r?
Your interpretation is right, and those minima are interesting in themselves. For example, when http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-a... was posted, the very first comment was about that.
"It almost looks as if the method doesn't (as presently written) produce explicit constants. Hopefully as the ideas become more widely understood, effective constants will be able to be extracted."
Ah thank you, the boundedness requirement makes sense.