The difference divides power of 2 2

Number Theory Level 1

If 2 x + 1 = y 2 2^x+1=y^2 for positive integers x , y x,y find x + y x+y (there is only one possibility)


The answer is 6.

Barack Clinton by Barack Clinton , 32, Switzerland

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3 solutions

Uahnbu Tran
Jun 4, 2015

2 x = ( y + 1 ) ( y − 1 ) ⇒ y + 1 = 2 a , y − 1 = 2 b ( a > b , a + b = x ) ⇒ 2 a − 2 b = 2 ⇒ 2 b ( 2 a − b − 1 ) = 2 ⇒ 2 b = 2 a n d 2 a − b − 1 = 1 { 2 }^{ x }=\left( y+1 \right) \left( y-1 \right) \\ \Rightarrow y+1=2^{ a },\quad y-1={ 2 }^{ b }\quad \left( a>b,\quad a+b=x \right) \\ \Rightarrow 2^{ a }-{ 2 }^{ b }=2\\ \Rightarrow 2^{ b }(2^{ a-b }-1)=2\\ \Rightarrow { 2 }^{ b }=2\quad and\quad 2^{ a-b }-1=1

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Ramiel To-ong
Jun 4, 2015

both x and y = 3 will only satisfy the given condition

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Noel Lo
Jun 3, 2015

8+1=9 where x=3, y=3 so x+y = 3+3 = 6.

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