Given x2 - Dy2 = 1:
(D-1)j2 + (j+1)2 = (x+y)2, where j = ((x+Dy) -1)/D
or
(D-1)j2 + (j-1)2 = (x+y)2, where j = ((x+Dy) +1)/D
Where two cases matter for finding integers, as j will be an integer if x = 1 mod D, for the first one, or x = -1 mod D for the second.
And gave this example back then:
With D=2, and x=17, y=12, you solve as 172 - 2(12)2 = 1, and going with the minus of the plus or minus:
j = ((17+2(12)-1)/2 = 20 is a solution giving:
202 + 212 = 292
So D-1 a perfect square relates to circles but otherwise gives non-circular ellipses.
And also I noted the rational solutions for x and y:
x = (D + t2)/(D - t2) and y = 2t/(D - t2)
Because that means you can get integer solutions for an ellipse, using those solutions, because of how those squares are distributed. But like with the first one, have always wondered about:
(D-1)j2 = (x+y)2 - (j+1)2 = (x + y + j + 1)(x + y - j - 1), j = ((x+Dy) - 1)/D
(D-1)j2 + (j+1)2 = (x+y)2, where j = ((x+Dy) -1)/D
or
(D-1)j2 + (j-1)2 = (x+y)2, where j = ((x+Dy) +1)/D
Where two cases matter for finding integers, as j will be an integer if x = 1 mod D, for the first one, or x = -1 mod D for the second.
And gave this example back then:
With D=2, and x=17, y=12, you solve as 172 - 2(12)2 = 1, and going with the minus of the plus or minus:
j = ((17+2(12)-1)/2 = 20 is a solution giving:
202 + 212 = 292
So D-1 a perfect square relates to circles but otherwise gives non-circular ellipses.
And also I noted the rational solutions for x and y:
x = (D + t2)/(D - t2) and y = 2t/(D - t2)
Because that means you can get integer solutions for an ellipse, using those solutions, because of how those squares are distributed. But like with the first one, have always wondered about:
(D-1)j2 = (x+y)2 - (j+1)2 = (x + y + j + 1)(x + y - j - 1), j = ((x+Dy) - 1)/D
Because you can force that to be all integers with rational solutions as all the denominators can be multiplied out. And I never checked it, to see if it ever factored D-1 non-trivially.
But have wondered, for years. Why will I not check it?
My guess? Is one of THE control equations for ALL integer factorizations. The other results from the other variant. That thing I suspect, helps control all integer factorization, across infinity. But maybe not.
May be key to integer factorization the way that x2 + y2 = 1 is key to trigonometry. Showing value of what I now call unary two conics equation. Maybe that thing controls ALL integer factorizations! To the math would be just about logic. For us? Would be so remarkable.
Maybe it just kind of scares me.
To just look at something that maybe controls so much, and wonder.
My guess? Is one of THE control equations for ALL integer factorizations. The other results from the other variant. That thing I suspect, helps control all integer factorization, across infinity. But maybe not.
May be key to integer factorization the way that x2 + y2 = 1 is key to trigonometry. Showing value of what I now call unary two conics equation. Maybe that thing controls ALL integer factorizations! To the math would be just about logic. For us? Would be so remarkable.
Maybe it just kind of scares me.
To just look at something that maybe controls so much, and wonder.
James Harris
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