please answer the question 3 and 4 below step by step r, s, u, and v from this s

Question

please answer the question 3 and 4 below step by step

r, s, u, and v from this solution in such way that the resulting new solution does have (x,y,t) = (t,z,w) = 1. Show that the equation x4 + 4y4 = z2, x 0, y 0, z 0 has no solutions. It may be helpful to reduce this to the case that x > 0, y > 0, z > 0, (x,y) = 1, and then by dividing by 4 (if necessary) to further reduce this to where x is odd. Show that there is no right triangle with integral sides whose area is a perfect square by showing that it suffices lo work with primitive triangles and with them, one is led to the equation of problem 3. (Hint: Do not use Theorem 5.2.)

Explanation / Answer

4.

I think you're going about it the right way, but are going to need some sort of descent argument to finish it off. It's probably easier not to apply the Pythagorean formula straight away, but first deduce facts about the sides.

OK, so we have a,b,c sides of a triangle and ab/2 is a square. Let us take c to be minimal - so in other words, we're looking for the counter-example with the smallest value of c possible. (This is possible because c is a natural number.)

Now a,b,c are pairwise coprime (if two of them share a prime factor p then it is easy to check that the third also shares this factor, so (a/p),(b/p),(c/p) form a smaller counter-example; this contradicts the minimality of c).

Also exactly one of a,b is even (otherwise c

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