4 a Give a proof by cases to show that the equation5x2 4y3 5

4. (a) Give a proof by cases to show that the equation5x^2+ 4y^3= 51 does not have solutions x, yZ+.

How do I get my cases?

Solution

the two cases are pretty simple,

in terms of y = ((515x^2)/4)^1/3
in terms of x = ((514y^3)/5)^1/2
now , we can see that both 4^1/3 and 5^1/2 are irrational numbers , and what we want is an integer , so we can directly say that there can be no such values