Is g ( x , y ) = f ( x 2 y + 1...

2nalfq8

2nalfq8

Answered

2022-07-16

Is g ( x , y ) = f ( x 2 y + 1 , y ) f ( x , y ) always an integer?

Answer & Explanation

Aryanna Caldwell

Aryanna Caldwell

Expert

2022-07-17Added 11 answers

If y is even, then
g ( x , y ) = x y ( 2 y + 1 ) 1 x 2 y + 1 + 1 x y 1 x + 1 = ( x y ( 2 y + 1 ) 1 ) ( x + 1 ) ( x 2 y + 1 + 1 ) ( x y 1 ) .
If y is odd, then
g ( x , y ) = x y ( 2 y + 1 ) 1 x 2 y + 1 1 x y 1 x 1 = ( x y ( 2 y + 1 ) 1 ) ( x 1 ) ( x 2 y + 1 1 ) ( x y 1 ) .
One has
x t 1 = d t Φ d ( x ) , x t + 1 = d 2 t d t Φ d ( x ) .
As a result, for example, using that gcd ( y , 2 y + 1 ) = 1 you can show that for y odd
g ( x , y ) = d y ( 2 y + 1 ) d y  or  2 y + 1 Φ d ( x ) .

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