(12C) Pell's Equation

[Gamo]

Pell's equation (also called the Pell–Fermat equation) is the equation of the form of

X^2 - nY^2 = 1

Where n is a given positive none square integer and integer solutions are sought for X and Y.

More information about this topic:
https://en.wikipedia.org/wiki/Pell%27s_equation

I personally make a note on this subject found on this MoHPC old forum and don't remember who posted this in the forum.
I found this very interesting and would like to share his clever program to solve this special equation.

Example: X^2 - (3)Y^2 = 1

3 R/S --> 2 X<>Y 1

Answer: X=2, Y=1

Remark: The list of the possible n is shown in the Wikipedia link.

Program: Pell's Equation

Code:


f 0
STO 0
SQRT
STO 1
INTG
STO 2
LSTx
FRAC
STO 3
1/x
INTG
STO 4
1
STO 5
STO 7
RCL 2
STO 6
RCL4
RCL 2
x
RCL 5
+
STO 2
RCL 1
/
RND
STO 7
RCL 3
1/x
FRAC
STO 3
1/x
INTG
STO 4
RCL 6
STO 5
RCL 2
ENTER
x
RCL 7
ENTER
x
RCL 0
x
-
1
-
X=0
GTO 51
GTO 16
RCL 7
RCL 2
GTO 00

Gamo