## JSH: Google problems, start of debate against series

Google Groups is on one of their delay things again so I can't see my new messages, so I can't reply through Google Groups, which is what I use to reply, to criticisms I see that have arisen.

So I'll make this post to note the problem, which hopefully will clear up in a few days, and to point out some things:

First, my infinite series was not previously known as you would have seen something like the following in a mathematical text on the subject and as so much has been written about Pell's Equation, I doubt you'd have missed it:
1. x^2 + Dy^2 = F

2. (x-Dy)^2 + D(x+y)^2 = F*(D+1)

3. ((1-D)x-2Dy)^2 + D(2x + (1-D)y)^2 = F*(D+1)^2

4. ((1-3D)x + (2D^2 - 4D)y)^2 + D((3-D)x + (1-3D)y)^2 = F*(D+1)^3
and that goes out to infinity. To get successive terms in the series you use the algebraic result that given:

u^2 + Dv^2 = C

it must be true that

(u-Dv)^2 + D(u+v)^2 = C*(D+1).

And where whenever the exponent of (D+1) is even, you can have a case where you just have a multiple of x and y, so you can solve for D, which defines possible values for F in terms of x or y.

Like with the third in the series:

((1-D)x-2Dy)^2 + D(2x + (1-D)y)^2 = F*(D+1)^2

you can consider

(1-D)x-2Dy = +/-(D+1)x or +/-(D+1)y

and

2x + (1-D)y = +/-(D+1)x or +/-(D+1)y

because trivially

(+/-(D+1)x)^2 + ((D+1)y)^2 = F*(D+1)^2

and solving for those possibilities gives a value for D. I got D=1, for the case of

(1-D)x-2Dy = (D+1)x and 2x + (1-D)y = (D+1)y.

So you can map the series as a super structure, where D is set at various levels.

One poster lobs a criticism that

u^2 + Dv^2 = C

requiring that

(u-Dv)^2 + D(u+v)^2 = C*(D+1)

is derivable from an equation known to Brahmagupta. Fine, so show that Brahmagupta derived that particular result and USED it to generate an infinite series.

Hey, you people missed the result right in front of you too, which may be why I'm seeing so many angry replies with insults.

Your chance of a lifetime was a post away and you missed it so anger can be a reaction.

And missing it against me when the opportunity to upstage me was there for days may really anger people who have argued with me or listened to and believed in other posters who argued with me as well.

That social stuff is powerful with a lot of people. Having people trust and believe in you, like when you're criticizing a "crackpot" can be powerful magic making a person feel all warm and fuzzy inside like they're actually SOMEBODY.

Getting upstaged pops the bubble. Reality is so annoying at times for some people, you know?

But you are now up against the worldwide community of people interested in mathematics who might want to explore that number theoretic super structure, and hatred is not an excuse for denigrating valuable mathematical ideas…not if you wish to be taken seriously later. Ever.

Students again I remind you of your futures. Angry sci.math posters are there because they're not good enough to go anywhere else, and I'm there because I'm an amateur who has been blocked out of usual circles, so I post to several math newsgroups including the one full of angry idiots.

But it is YOUR future. Angry losers on sci.math will probably keep ripping on this result just because…well, because they're losers, who you'll notice could not see that infinite series right in front of them for days.

They were too busy still ripping on me.

Don't make it personal. It's math. You make it personal and you lose.

So you all lost round one. Easy mathematical fame escaped you with the find of the series, so now it's up to you what you do next.

Continue to listen to angry idiots, or do the math.