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Serbia JBMO TST
2023 Serbia JBMO TST
4
4
Part of
2023 Serbia JBMO TST
Problems
(1)
Seemingly ugly NT with 4 conditions
Source: Serbia JBMO TST 2023 P4
5/28/2023
Find all triples
(
k
,
m
,
n
)
(k, m, n)
(
k
,
m
,
n
)
of positive integers such that
m
m
m
is a prime and:(1)
k
n
kn
kn
is a perfect square;(2)
k
(
k
−
1
)
2
+
n
\frac{k(k-1)}{2}+n
2
k
(
k
−
1
)
+
n
is a fourth power of a prime;(3)
k
−
m
2
=
p
k-m^2=p
k
−
m
2
=
p
where
p
p
p
is a prime;(4)
n
+
2
m
2
=
p
4
\frac{n+2}{m^2}=p^4
m
2
n
+
2
=
p
4
.
number theory