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2022 Kurschak Competition
2
2
Part of
2022 Kurschak Competition
Problems
(1)
Solutions to |px^2-qy^2|=1
Source: Kürschák József Competition 2022/2
10/7/2022
Let
p
p
p
and
q
q
q
be prime numbers of the form
4
k
+
3
4k+3
4
k
+
3
. Suppose that there exist integers
x
x
x
and
y
y
y
such that
x
2
−
p
q
y
2
=
1
x^2-pqy^2=1
x
2
−
pq
y
2
=
1
. Prove that there exist positive integers
a
a
a
and
b
b
b
such that
∣
p
a
2
−
q
b
2
∣
=
1
|pa^2-qb^2|=1
∣
p
a
2
−
q
b
2
∣
=
1
.
number theory
prime numbers
square