(35S) Pell's Equation Programme
06-07-2014, 09:17 AM (This post was last modified: 12-25-2018 10:41 AM by Gerald H.)
Post: #1
 Gerald H Senior Member Posts: 1,627 Joined: May 2014
(35S) Pell's Equation Programme
The programme finds the primitive integer solution of Pell’s equation
x^2 – D * y^2 = ± 1
for integer input D and 1 or -1 for solution of the +1 or -1 case respectively.

Code:
1    LBL P 2    STO K 3    1 4    STO F 5    REGZ►A 6    ENTER 7    √x 8    IP 9    STO B 10    STO D 11    STO H 12    x^2 13    - 14    STO I 15    x=0? 16    RTN 17    2 18    RCL* B 19    RCL/ I 20    IP 21    STO C 22    STO G 23    RCL* B 24    1 25    + 26    STO E 27    RCL I 28    1 29    x≠y? 30    GTO P037 31    RCL K 32    x>0? 33    GTO P082 34    RCL A 35    [D,F,-1] 36    RTN 37    0 38    STO J 39    RCL J 40    NOT 41    STO J 42    RCL C 43    RCL* I 44    RCL- H 45    STO H 46    x^2 47    +/- 48    RCL+ A 49    RCL/ I 50    IP 51    STO I 52    1 53    x≠y? 54    GTO P067 55    RCL K 56    x<0? 57    GTO P062 58    RCL J 59    x≠0? 60    GTO P082 61    GTO P067 62    RCL J 63    x=0? 64    GTO P082 65    0 66    RTN 67    RCL B 68    RCL+ H 69    RCL/ I 70    IP 71    STO C 72    RCL* E 73    RCL+ D 74    x<>E 75    STO D 76    RCL C 77    RCL* G 78    RCL+ F 79    x<>G 80    STO F 81    GTO P039 82    RCL A 83    [E,G,K] 84    RTN

For input 13 & 1 the programme returns [649, 180, 1] & indeed
649^2 – 13 * 180^2 = 1
Similarly input 13 & -1 returns [18, 5, -1] &
18^2 – 13 * 5^2 = -1
Input 7 & -1 returns 0, indicating that
x^2 – 7 * y^2 = -1
has no integer solutions for x & y.
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