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Re: x^2 - Ay^2 =1
Posted:
Jun 14, 2007 4:25 AM
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continued...
Hi,
we will give to all the polinomials formulas that
give the primitive of Pell's equation x^2-Ay^2=1
*****************************************
(7)
If A=(n^2+2)/9, then y=3n, and x=n^2+1
(with n>=13, and n^2+2=0 mod.9)
n=(13,14,22,23,31,32,...,)
A=(19,22,54,59,107,114,...,)
Y=(39,42,66,69,93,96,...,)
X=(170,197,485,530,962,1025,...,)
*****************************************
(8)
If A=4n^2+4n-3, y=2n^2+2n, x=4n^3+6n^2-1
(with n>=1)
A=(5,21,45,77,117,165,221,285,...,)
Y=(4,12,24,40,60,84,112,144,...,)
X=(9,55,161,351,649,1079,1665,2431,...,)
*****************************************
(9)
If A=(n^2-1)/24^2, then y=24, and x=n
(with n>=127 and n^2-1=0 mod.24^2)
n=(127,161,287,289,415,449,...,)
A=(28,45,143,145,299,350,...,)
Y=(24,24,24,24,24,24,...,)
X=(127,161,287,289,415,449,...,)
*****************************************
(10)
If A=(n^2+1)/13^2, y=26n, x=2n^2+1
(with n>=70 and n^2+1=0 mod.13^2)
n=(70,99,239,268,408,437,...,)
A=(29,58,338,425,985,1130,...,)
Y=(1820,2574,6214,6968,10608,11362,...,)
X=(9801,19603,114243,143649,332929,381939.,)
********************************************
(11)
If A=(n^2-2)/7^2, y=7n, x=n^2-1
(with n>10 and n^2-2=0 mod.7^2)
n=39,59,88,108,137,...,)
A=(31,71,158,238,383,...,)
Y=(273,413,616,756,959,...,)
X=(1520,3480,7743,11663,18768,...,)
***********************************
(12)
If A=(n^2-1)/3^2, y=3, x=n
(with n>=8 and n^2-1=0 mod.3^2)
n=(8,10,17,19,26,28,35,37,44,...,)
A=(7,11,32,40,75,87,136,152,215,...,)
Y=(3,3,3,3,3,3,3,3,3,...,)
X=(8,10,17,19,26,28,35,37,44,...,)
*************************************
continued....
Regards
Vincenzo Librandi
vincenzo.librandweoz@alice.it
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