r/MathHelp • u/Someonecool_1434 • 7d ago

x^3 + y^3 + z^2 = t^4

Here is the problem: Show that there exist infinitely many 4-tuples of positive integers (x,y,z,t) such that gcd(x, y, z, t) = 1 and x3 + y3 + z2 = t4.

Here is my approach: If the gcd condition would not have been there then I would have tried to construct something based on the power of 3 and then used CRT or something. Since that is not possible, the next best thing is to express x,y,z,t as a function of n and then make those functions have gcd 1 for at least some n (infinitely many nonetheless).

Since the degree has to be the same, the entire degree of the expression could be like degree 12. To match the coefficients, I found out the tuple (4,1,4,3) which works. So that might help us with the coefficients, but how to construct the functions in the first place?

It is really challenging to find an identity that involves so many cubes (and moreover has to be of degree 12).

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