ELEMENTARY NUMBER THEORY PROBLEM SHEET 9
To be handed in at the start of your tutorial in the week beginning
Mar 27th
Exercise 1. Consider a right angled triangle with side lengths x, y, z (z is the
length of the hypotenuse) and denote the radius of the inscribed circle (or incircle)
by r (see diagram below).
By considering the area of the triangle, show that we have
1
1
xy = r(x + y + z)
2
2
Show that, if x, y and z are integers, r is also an integer.
z
x
r
y
Exercise 2.
(1) Fix a positive integer x0 . Show that are only finitely many
integer solutions (y, z) to the equation x20 + y 2 = z 2 . (Hint: consider the
equation x20 = (z + y)(z − y)).
(2) Find three different Pythagorean triples (not necessarily primitive) of the
form (16, y, z).
(3) Find all primitive Pythagorean triples of the form (40, y, z).
Exercise 3. Find all Pythagorean triangles whose areas are equal to their perimeter. (Hint: use the description of Pythagorean triples from lectures)
Exercise 4. Suppose x, y, z are non-zero integers with y 2 = x3 +xz 4 , and moreover
suppose that x, z are coprime.
(1) Show that x and x2 + z 4 are coprime, and deduce that x and x2 + z 4 are
both squares of integers.
(2) Using the fact that the equation a4 + b4 = c2 has no non-zero integer
solutions, show that y 2 = x3 + xz 4 has no non-zero integer solutions with
x, z coprime.
(3) (*) Show that y 2 = x3 + xz 4 has no non-zero integer solutions.
Exercise 5. Consider the unit circle C, with equation x2 + y 2 = 1.
(1) Let m be a rational number, and let L be the line passing through (−1, 0)
with slope m. What is the other intersection point between L and C?
Date: Monday 13th March, 2017.
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2
ELEMENTARY NUMBER THEORY PROBLEM SHEET 9
(2) Show that taking intersections between L and C as in the previous part
(and letting the rational slope m vary) gives all rational solutions to the
equation x2 + y 2 = 1.
(3) (*) Use the above two parts to give a new proof of the description from
lectures of all primitive Pythagorean triples.
Exercise 6. (*) Suppose x, y, z are positive integers with x4 − y 4 = 2z 2 , and
moreover suppose that x, y are coprime.
(1) Show that x and y are both odd and gcd(x2 − y 2 , x2 + y 2 ) = 2.
(2) Show that there exist integers a, d such that x2 −y 2 = d2 and x2 +y 2 = 2a2 .
(Hint: first show that (x2 − y 2 ) and (x2 + y 2 )/2 are coprime)
(3) Using the previous part, show that there exist integers b, c such that x+y =
2b2 and x − y = 2c2 .
(4) Show that the integers a, b, c satisfy a2 = b4 + c4 and deduce that there are
no integer solutions to the equation x4 − y 4 = 2z 2 .
Exercise 7. (**)
(1) Show that the equation x4 − y 4 = z 2 has no positive integer solutions.
(Hint: you can use a modification of the argument applied in lectures to
the equation x4 + y 4 = z 2 ).
(2) Using the first part, show that the area of a right angled triangle with
integer length sides cannot be the square of an integer. Deduce that there
is no right angled triangle with area 1 and rational length sides.
(*) denotes a trickier exercise for those who are interested — it is not essential
for the course. (**) is a longer and/or trickier exercise.