Prove that the equation 2x−1=y2 has no integer solutions where x>1.

Wanda Kane

Wanda Kane

Answered

2022-01-15

Prove that the equation
2x1=y2 has no integer solutions where x>1.

Answer & Explanation

macalpinee3

macalpinee3

Expert

2022-01-16Added 29 answers

y=2x1 has a root for x=0, because its asymptotic at y=1, and is continually increasing. However, y2=2x1 tells a different story. It still has a root at x=0, but its parabolic in nature in the horizontal direction.
The reason your function has no roots because of your domain of x>1dydx>0,for x>1 and at x=1,y=1. Hence, what can you conclude about the nature of the function and its position in a cartesian plane after x>1. Will it every hit the x-axis?

movingsupplyw1

movingsupplyw1

Expert

2022-01-17Added 30 answers

The point of a mathematical proof is not to be mathematical
alenahelenash

alenahelenash

Expert

2022-01-24Added 366 answers

Note that every square number is either 0mod4 or 1mod4. It follows that for x>2, we have that 2x1=3mod4 and hence, cannot be a square. So if there is a solution it must happen when x2. We can manually check that the only remaining potential solution of x=2 gives no integer solutions for y.

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