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National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Kyiv City MO
2021 Kyiv City MO
2021 Kyiv City MO Round 1
11.5
11.5
Part of
2021 Kyiv City MO Round 1
Problems
(1)
Sum of powers
Source: Kyiv City MO 2021 Round 1, Problem 11.5
12/21/2023
For positive integers
m
,
n
m, n
m
,
n
define the function
f
n
(
m
)
=
1
2
n
+
2
2
n
+
3
2
n
+
…
+
m
2
n
f_n(m) = 1^{2n} + 2^{2n} + 3^{2n} + \ldots +m^{2n}
f
n
(
m
)
=
1
2
n
+
2
2
n
+
3
2
n
+
…
+
m
2
n
. Prove that there are only finitely many pairs of positive integers
(
a
,
b
)
(a, b)
(
a
,
b
)
such that
f
n
(
a
)
+
f
n
(
b
)
f_n(a) + f_n(b)
f
n
(
a
)
+
f
n
(
b
)
is a prime number.Proposed by Nazar Serdyuk
number theory
prime numbers