(a^2 + b^2 + c^2 + d^2)(e^2 + f^2 + g^2 + h^2) as a sum of 7 squares

 

(a^2+b^2+c^2+d^2) \,(e^2+f^2+g^2+h^2)

= a^2 e^2+a^2 f^2+a^2 g^2+a^2 h^2+b^2 e^2+b^2 f^2+b^2 g^2+b^2 h^2+c^2 e^2+c^2 f^2+c^2 g^2
     + \; c^2 h^2+d^2 e^2+d^2 f^2+d^2 g^2+d^2 h^2

= (ae + bf + cg + dh)^2 + (af - be)^2 + (ag - ce)^2 + (ah - de)^2 + (bg - cf)^2
     + \; (bh - df)^2 + (ch - dg)^2

 
for example,
 

(1)   1^2 + 6^2 + 7^2 + 20^2 \; = \; 2^2 + 4^2 + 5^2 + 21^2 \; = \; 486
(2)   1^2 + 5^2 + 12^2 + 18^2 \; = \; 2^2 + 3^2 + 9^2 + 20^2 \; = \; 494

(1^2 + 6^2 + 7^2 + 20^2) \,(1^2 + 5^2 + 12^2 + 18^2) = 486 \times 494 = 240084
(1^2 + 6^2 + 7^2 + 20^2) \,(2^2 + 3^2 + 9^2 + 20^2) = 486 \times 494 = 240084
(1^2 + 6^2 + 7^2 + 20^2) \,(2^2 + 4^2 + 5^2 + 21^2) = 486^2
(2^2 + 4^2 + 5^2 + 21^2) \,(2^2 + 3^2 + 9^2 + 20^2) = 486 \times 494 = 240084
(1^2 + 5^2 + 12^2 + 18^2) \,(2^2 + 3^2 + 9^2 + 20^2) = 494^2

 
(1^2 + 6^2 + 7^2 + 20^2) \,(2^2 + 4^2 + 5^2 + 21^2)
= \; 2^2 + 8^2 + 9^2 + 19^2 + 46^2 + 47^2 + 481^2
= \; 486^2

(1^2 + 5^2 + 12^2 + 18^2) \,(2^2 + 3^2 + 9^2 + 20^2)
= \; 7^2 + 9^2 + 15^2 + 16^2 + 46^2 + 78^2 + 485^2
= \; 494^2

 
(1^2 + 6^2 + 7^2 + 20^2) \,(1^2 + 5^2 + 12^2 + 18^2)
= \; 1^2 + 2^2 + 5^2 + 8^2 + 37^2 + 114^2 + 475^2
= \; 240084

(1^2 + 6^2 + 7^2 + 20^2) \,(2^2 + 3^2 + 9^2 + 20^2)
= \; 5^2 + 9^2 + 20^2 + 33^2 + 40^2 + 60^2 + 483^2
= \; 240084

By eliminating   5^2,   we get,
1^2+2^2+8^2+37^2+114^2+475^2 = 9^2+20^2+33^2+40^2+60^2+483^2 = 240059
240059   is a prime number.

 
(2^2 + 4^2 + 5^2 + 21^2) \,(2^2 + 3^2 + 9^2 + 20^2)
= \; 2^2 + 2^2 + 8^2 + 17^2 + 21^2 + 89^2 + 481^2
= \; 240084

240084 \; = \; 1^2 + 2^2 + 5^2 + 8^2 + 37^2 + 114^2 + 475^2
240084 \; = \; 5^2 + 9^2 + 20^2 + 33^2 + 40^2 + 60^2 + 483^2
240084 \; = \; 2^2 + 2^2 + 8^2 + 17^2 + 21^2 + 89^2 + 481^2

 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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