Here is a fun little result, which is a mathematical absolute, with all positive integers:
x + y + j + 1 = n2 or 2n2
if x = 1 mod D, x2 - Dy2 = 1, and j = (x+Dy-1)/D
For example, 82 - 7*32 = 1, so x = 8, and 8 mod 7 = 1.
j = (8 + 7*3 - 1)/7 = (7 + 7*3)/7 = 4
8 + 3 + 4 + 1 = 16 = 42
That's a simple example to make this post easy to write for me, but the result is true over infinity.
So whenever x = 1 mod D, with x2 - Dy2 = 1, then these rules are forced, absolutely.
Seem easy or trivial?
It's another key piece of the explanation for an ancient math mystery.
James Harris
Global resource of innovative mathematical ideas. Discovery for the 21st century. Abstract reductionism realized. And modular rules.
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Saturday, May 30, 2015
Sunday, May 10, 2015
Part of the key to an ancient math mystery
Known since antiquity: x2 - Dy2 = 1
The smallest integer solution by absolute value for x and y, not zero is called the fundamental solution. Sometimes it can be quite large, but why?
For example, for D = 61, the fundamental solution is quite large:
17663190492 - 61(226153980)2 = 1
Part of the key to the answer--hidden in simple math for millennia:
(D-1)j2 + (j - 1)2 = (x+y)2,
where j = (x+Dy+1)/D, so our x, gives: j = 255110030.
60*2551100302 + 2551100292 = 19924730292
A connection like no other, an ellipse to a hyperbola--an answer? Part of it.
Here's another example, for D = 313, which I found in a Wikipedia article:
321881208291348492 - 313(1819380158564160)2 = 1
so:
j = 1922217605302610
312*19222176053026102 + 19222176053026092 = 340075009876990092
That one is too big for my pc calculator, though I could program something to check it, I'll use another trick.
I know that when x = -1 mod D that x+y+j-1 must be a perfect square or twice one.
That's an absolute by the way.
And yup, x+y+j-1 = 2*1340330532 so it works! Oh yeah, same thing works above, which is another clue.
If x = -1 mod D and x2 - Dy2 = 1, with all nonzero positive integers, then these equations apply, across infinity. So oh yeah, it's an infinity result. I get a kick out of those things, try now to note when I have one.
Oh yeah, remembered I could use Wolfram Alpha and everything checked out ok there.
Who knew? Ellipses and hyperbolas, like a freaky number family?
Of course I have the full answer. It's ok. To me it's just nice to know, but I like having answers to mysteries.
James Harris
The smallest integer solution by absolute value for x and y, not zero is called the fundamental solution. Sometimes it can be quite large, but why?
For example, for D = 61, the fundamental solution is quite large:
17663190492 - 61(226153980)2 = 1
Part of the key to the answer--hidden in simple math for millennia:
(D-1)j2 + (j - 1)2 = (x+y)2,
where j = (x+Dy+1)/D, so our x, gives: j = 255110030.
60*2551100302 + 2551100292 = 19924730292
A connection like no other, an ellipse to a hyperbola--an answer? Part of it.
Here's another example, for D = 313, which I found in a Wikipedia article:
321881208291348492 - 313(1819380158564160)2 = 1
so:
j = 1922217605302610
312*19222176053026102 + 19222176053026092 = 340075009876990092
That one is too big for my pc calculator, though I could program something to check it, I'll use another trick.
I know that when x = -1 mod D that x+y+j-1 must be a perfect square or twice one.
That's an absolute by the way.
And yup, x+y+j-1 = 2*1340330532 so it works! Oh yeah, same thing works above, which is another clue.
If x = -1 mod D and x2 - Dy2 = 1, with all nonzero positive integers, then these equations apply, across infinity. So oh yeah, it's an infinity result. I get a kick out of those things, try now to note when I have one.
Oh yeah, remembered I could use Wolfram Alpha and everything checked out ok there.
Who knew? Ellipses and hyperbolas, like a freaky number family?
Of course I have the full answer. It's ok. To me it's just nice to know, but I like having answers to mysteries.
James Harris
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