Google Groups no longer supports new Usenet posts or subscriptions. Historical content remains viewable.
Dismiss

Squares represented in 7 ways as a sum of 3 squares

0 views
Skip to first unread message

M. A. Fajjal

unread,
Dec 12, 2008, 10:42:50 PM12/12/08
to
Squares represented in 7 ways as a sum of 3 squares

Hi

D^2 =A1^2+B1^2+C1^2 = A2^2+B2^2+C2^2 = A3^2+B3^2+C3^2 = A4^2+B4^2+C4^2 = A5^2+B5^2+C5^2 = A6^2+B6^2+C6^2 = A7^2+B7^2+C7^2


a = (b^2+c^2+d^2-p^2)
m = b*p
n = c*p
u = d*p


A1 = (a^2+m^2-n^2-u^2)
B1 = (2*a*n+2*m*u)
C1 = (2*a*u-2*m*n)

A2 = p^2*(b^2+c^2+d^2-p^2)
B2 = (b^2+c^2+d^2)*(b^2+c^2+d^2-p^2)
C2 = (b^2+c^2+d^2)*p^2

A3 = (n^2+m^2-a^2-u^2)
B3 = (2*a*n+2*m*u)
C3 = (2*n*u-2*m*a)

A4 = (u^2+m^2-n^2-a^2)
B4 = (2*u*n+2*m*a)
C4 = (2*a*u-2*m*n)

A5 = (n^2+m^2-a^2-u^2)
B5 = (2*u*n+2*m*a)
C5 = (2*n*a-2*m*u)

A6 = (a^2+m^2-n^2-u^2)
B6 = (-2*a*n+2*m*u)
C6 = (-2*a*u-2*m*n)

A7 = (u^2+m^2-n^2-a^2)
B7 = (2*u*n-2*m*a)
C7 = (-2*a*u-2*m*n)

D = (a^2+m^2+n^2+u^2)

where b, c and d are different integers

Exapmle b=1, c=2, d=3, p=3

151^2 = 83^2+114^2+54^2 = 45^2+70^2+126^2 = 61^2+114^2+78^2 = 29^2+138^2+54^2 = 61^2+138^2+6^2 = 83^2+6^2+126^2 = 29^2+78^2+126^2

M. A. Fajjal

unread,
Dec 13, 2008, 3:33:25 AM12/13/08
to
In general 151^2 can be represented in 19 ways

151^2
= 83^2+114^2+54^2
= 45^2+70^2+126^2
= 61^2+114^2+78^2
= 29^2+138^2+54^2
= 61^2+138^2+6^2
= 83^2+6^2+126^2
= 29^2+78^2+126^2

= 90^2+99^2+70^2
= 90^2+110^2+51^2
= 99^2+110^2+30^2
= 99^2+114^2+2^2
= 99^2+106^2+42^2
= 54^2+141^2+2^2
= 93^2+106^2+54^2
= 114^2+34^2+93^2
= 141^2+34^2+42^2
= 138^2+34^2+51^2
= 147^2+34^2+6^2
= 142^2+51^2+6^2

0 new messages