When (x+y) and (x^2 + y^2) are given

 
 
x^3 + y^3 = (x^2 + y^2)(x + y) - 1/2 \; ((x + y)^2 - (x^2 + y^2))(x + y)
x^4 + y^4 = (x^3 + y^3)(x + y) - 1/2 \; ((x + y)^2 - (x^2 + y^2))(x^2 + y^2)
x^5 + y^5 = (x^4 + y^4)(x + y) - 1/2 \; ((x + y)^2 - (x^2 + y^2))(x^3 + y^3)
x^6 + y^6 = (x^5 + y^5)(x + y) - 1/2 \; ((x + y)^2 - (x^2 + y^2))(x^4 + y^4)
x^7 + y^7 = (x^6 + y^6)(x + y) - 1/2 \; ((x + y)^2 - (x^2 + y^2))(x^5 + y^5)
x^8 + y^8 = (x^7 + y^7)(x + y) - 1/2 \; ((x + y)^2 - (x^2 + y^2))(x^6 + y^6)
x^9 + y^9 = (x^8 + y^8)(x + y) - 1/2 \; ((x + y)^2 - (x^2 + y^2))(x^7 + y^7)
………………..
 
 

If   x + y = a    and    x^2 + y^2 = b,   then
 

x^3 + y^3 = 1/2 \; (3 \, a \, b - a^3)
x^4 + y^4 = 1/2 \; (-a^4 + 2 \, a^2 \, b + b^2)
x^5 + y^5 = 1/4 \; a \, (5 \, b^2 - a^4)
x^6 + y^6 = 1/4 \; b \, (-3 \, a^4 + 6 \, a^2 \, b + b^2)
x^7 + y^7 = 1/8 \; a \, (a^6 - 7 \, a^4 \, b + 7 \, a^2 \, b^2 + 7 \, b^3)
x^8 + y^8 = 1/8 \; (a^8 - 4 \, a^6 \, b - 2 \, a^4 \, b^2 + 12 \, a^2 \, b^3 + b^4)
x^9 + y^9 = 1/16 \; a \, (a^8 - 18 \, a^4 \, b^2 + 24 \, a^2 \, b^3 + 9 \, b^4)

 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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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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