Download MA3060 Exercise Set 2 Q1. In any primitive Pythagorean triple (x, y

MA3060
Exercise Set 2
Q1. In any primitive Pythagorean triple (x, y, z) show that exactly one of
the x, y, z is divisible by 5.
Q2. In each of the following cases, prove that there are infinitely many
triangles whose side lengths a, b, c are relatively prime integers, and whose angles
satisfy the required property. In each case write a concrete formula for such integer
triples (a, b, c):
a) ∠A = 60◦ ;
b) ∠A = 120◦ ;
c) ∠A = 2∠B.
Q3. Consider the Diophantine equation 3x2 + 7y 2 = 3z 2 . Find all integral
solutions (x, y, z) up to scaling.
Q4. Consider the Diophantine equation Ax2 + By 2 = Cz 2 .
a) Give an example of triples (A, B, C) so that the equation above has no
solutions in R \ {0}.
b) Prove that 3x2 + 5y 2 = z 2 has no solutions in Z \ {0}, even though it does
have solutions in R \ {0}.
Q5. (Fermat) Consider the equation x4 + y 4 = z 2 . Prove that it has no
integer solutions in Z \ {0}. You may follow the strategy known as infinite descent:
(i) Assuming that integer solutions exist, consider the one with the smallest
positive z.
(ii) Apply the formula for Pythagorean triples twice, thus writing x, y, z in
terms of integer parameters s and t.
(iii) Conclude that s and t are also the squares of some integers, and hence
derive a contradiction with part (i).
Q6. Check that the Diophantine equation x3 + y 3 + z 3 = t3 has an integer
solution (x, y, z, t) = (3, 4, 5, 6) and prove that it has infinitely many solutions.
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