Question

Find all solutions x,y∈Z to the following Diophantine equations: (a) x²=y³. (b) x²−x=y³. (c) x²=y⁴−77. (d) x³=4y²+4y−3.

Answer 1

The solutions to the given Diophantine equations in integers are provided.

Step-by-step explanation:

Diophantine Equations

1. (a) The Diophantine equation x²=y³ represents a special case of the Mordell equation. It is also known as an elliptic curve. The solutions to this equation in integers (x,y) are (0,0), (1,1), and (-1,1).
2. (b) The Diophantine equation x²-x=y³ can be rewritten as x(x-1)=y³. The only solutions to this equation in integers are (0,0) and (1,1).
3. (c) The Diophantine equation x²=y⁴-77 can be rewritten as x²+77=y⁴. There are no integer solutions to this equation.
4. (d) The Diophantine equation x³=4y²+4y-3 can be rewritten as x³-4y²-4y+3=0. The only integer solution to this equation is (1,0).