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potm 2 .pdf


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2016 September POTM
Solution by Benjamin Thomas
Prove: a, b ∈ Z+ and ab + 1 | a2 + b2 =⇒

a2 +b2
ab+1

= k 2 for k ∈ Z

Our goal is to express a, b in terms of k and show that k must be a perfect square
for a, b to be integers.
a 2 + b2
= k =⇒ a2 + b2 − kab = k
We have
ab + 1
We notice that the left hand side is a quadratic form


1 −k
a
a b
−k 1
b
We will refer to a as the vector (a, b) and A as the matrix associated with the
quadratic form.

We now make a change of variable a = P x where P is the matrix that orthogonally diagonalizes A.
We find that
!
√1
√1

2
P = √12
√1
2

2

leading to the quadratic form


1−k
0
x
x y
=⇒ (1 − k)x2 + (1 + k)y 2 = k
0
1+k
y
Solving this yields
r
x=

k − (1 + k)t2
1−k

y=t
And finally
s

k − (1 + k)t2
t
−√
2(1 − k)
2

s

t
k − (1 + k)t2
+√
2(1 − k)
2

a=

b=
Where t ∈ R

1


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