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National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2019 Korea Junior Math Olympiad.
3
3
Part of
2019 Korea Junior Math Olympiad.
Problems
(1)
Find all pairs of prime numbers (p, q)
Source: KJMO 2019 p3
1/8/2021
Find all pairs of prime numbers
p
,
q
(
p
≤
q
)
p,\,q(p\le q)
p
,
q
(
p
≤
q
)
satisfying the following condition: There exists a natural number
n
n
n
such that
2
n
+
3
n
+
⋯
+
(
2
p
q
−
1
)
n
2^{n}+3^{n}+\cdots+(2pq-1)^{n}
2
n
+
3
n
+
⋯
+
(
2
pq
−
1
)
n
is a multiple of
2
p
q
2pq
2
pq
.
number theory
prime numbers
KJMO