Any integer n; n = i^2 + j^2 – 5k^2

 

For any integer   n   there exists integers   i, \; j,   and   k   such that

n \; = \; i^2 \; + \; j^2 \; - \; 5 \,k^2

(m - a_1)^2 + (2 \, m - b_1)^2 - 5(m - c_1)^2 \; = \; 2 \, m \; = \; n
(m - a_2)^2 + (2 \, m - b_2)^2 - 5(m - c_2)^2 \; = \; 2 \, m \; + \; 1 \; = \; n

a_1 \; = \; 2 \,(5 \, k^2 + 5 \, k + 1)
b_1 \; = \; 20 \, k^2 + 10 \,(2 - t) \,k + (6 - 5 \, t)
c_1 \; = \; 10 \, k^2 + 2 \,(5 - 2 \, t) \,k + (3 - 2 \, t)

a_2 \; = \; 10 \, k^2 \; - \; 1
b_2 \; = \; 10 \,k \,(2 \,k + t)
c_2 \; = \; 2 \,k \,(5 \,k + 2 \,t)

with   t = \pm 1   and   k   any integer.

 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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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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