### Sunday, February 15, 2009

## General solution to Pell's Equation

General solution to Pell's Equation

Given

x^2 - Dy^2 = 1

I have proven:

y = [+/-2Dv/(f_1 - f_2*v^2 - 2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v)]/(D-1)

and

x = +/-(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v) - [+/-2Dv/(f_1 - f_2*v^2 - 2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v)]/(D-1)

where f_1*f_2 = D-1, and the f's are non-zero integer factors, while v is a free variable.

As Pell's Equation is normally considered in integers as a Diophantine equation note that you find rational v such that x and y are integers, which gives the 'why' of Pell's Equation. For instance, for D=2, f_1*f_2 = 1, so I have:

y = [+/-4v/(1 - v^2 - 2v) +/- 1 -/+(1 + v^2)/(1 - v^2 - 2v)]

and

x = +/-(1 + v^2)/(1 - v^2 - 2v) - [+/-4v/(1 - v^2 - 2v) +/- 1 -/+(1 + v^2)/(1 - v^2 - 2v)]

where I notice an easy case to give integer solutions with v = -2, as I have then:

y = [-/+8/(1 - 4 + 4) +/- 1 -/+(1 + 4)/(1 - 4 + 4)] = [-/+8 +/- 1 -/+ 5]

and

x = +/-(1 + 4)/(1 - 4 + 4) - [-/+8/(1 - 4 + 4) +/- 1 -/+(1 + 4)/(1 - 4 + 4)] = +/-5 - [-/+8 +/- 1 -/+ 5]

So I get several solutions with that choice.

For instance, y = 8 - 1 + 5 = 12 is a solution, with x = 5 + 12 = 17, as 17^2 - 2(12)^2 = 1.

Derivation of the result was done using research found by doors opened by my Quadratic Diophantine Theorem, my use of it against Pell's Equation, and my research result linking Pell's Equation to discrete ellipses and Pythagorean Triples.

The value of basic research is shown by how somewhat disparate results came together to answer a classical problem that is over 2000 years old in an entirely new way, shattering beliefs I have come across within the modern mathematical community that there are not new answers to be found in seemingly well-worked areas.

Mathematics is an infinite subject. That is a good thing. Human beings limit mathematics out of intellectual ignorance.

Discovery never ends unless people decide for ignorance over knowledge.

Given

x^2 - Dy^2 = 1

I have proven:

y = [+/-2Dv/(f_1 - f_2*v^2 - 2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v)]/(D-1)

and

x = +/-(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v) - [+/-2Dv/(f_1 - f_2*v^2 - 2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v)]/(D-1)

where f_1*f_2 = D-1, and the f's are non-zero integer factors, while v is a free variable.

As Pell's Equation is normally considered in integers as a Diophantine equation note that you find rational v such that x and y are integers, which gives the 'why' of Pell's Equation. For instance, for D=2, f_1*f_2 = 1, so I have:

y = [+/-4v/(1 - v^2 - 2v) +/- 1 -/+(1 + v^2)/(1 - v^2 - 2v)]

and

x = +/-(1 + v^2)/(1 - v^2 - 2v) - [+/-4v/(1 - v^2 - 2v) +/- 1 -/+(1 + v^2)/(1 - v^2 - 2v)]

where I notice an easy case to give integer solutions with v = -2, as I have then:

y = [-/+8/(1 - 4 + 4) +/- 1 -/+(1 + 4)/(1 - 4 + 4)] = [-/+8 +/- 1 -/+ 5]

and

x = +/-(1 + 4)/(1 - 4 + 4) - [-/+8/(1 - 4 + 4) +/- 1 -/+(1 + 4)/(1 - 4 + 4)] = +/-5 - [-/+8 +/- 1 -/+ 5]

So I get several solutions with that choice.

For instance, y = 8 - 1 + 5 = 12 is a solution, with x = 5 + 12 = 17, as 17^2 - 2(12)^2 = 1.

Derivation of the result was done using research found by doors opened by my Quadratic Diophantine Theorem, my use of it against Pell's Equation, and my research result linking Pell's Equation to discrete ellipses and Pythagorean Triples.

The value of basic research is shown by how somewhat disparate results came together to answer a classical problem that is over 2000 years old in an entirely new way, shattering beliefs I have come across within the modern mathematical community that there are not new answers to be found in seemingly well-worked areas.

Mathematics is an infinite subject. That is a good thing. Human beings limit mathematics out of intellectual ignorance.

Discovery never ends unless people decide for ignorance over knowledge.