The equation a2+b2 = c2 has lots of solutions in positive integers, while the equation a3 + b3 = c3 has no solutions in positive integers. This exercise asks you to look for solutions to the equation
a3 + b3 = c2 ---1
in integers c ≥ b ≥ a ≥ 1.
(a) The equation (1) has the solution (a, b, c) = (2, 2, 4). Find three more solutions in positive integers. [Hint. Look for solutions of the form (a, b, c) = (xz, yz, z2). Not every choice of x, y, z will work, of course, so you’ll need to figure out which ones do work.]
(b) If (A,B,C) is a solution to (1) and n is any integer, show that (n2A, n2B, n3C) is also a solution to (1). We will say that a solution (a, b, c) to (1) is primitive if it does not look like (n2A, n2B, n3C) for any n ≥ 2.
(c) Write down four different primitive solutions to (1). [That is, redo (a) using only primitive solutions.]
(d) The solution (2, 2, 4) has a = b. Find all primitive solutions that have a = b.
(e) Find a primitive solution to (1) that has a > 10000.
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