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National and Regional Contests
Brazil Contests
Brazil National Olympiad
2012 Brazil National Olympiad
4
4
Part of
2012 Brazil National Olympiad
Problems
(1)
Sums of powers with prime exponents, summing a square
Source: Problem 4, Brazilian MO, 2012
11/28/2012
There exists some integers
n
,
a
1
,
a
2
,
…
,
a
2012
n,a_1,a_2,\ldots,a_{2012}
n
,
a
1
,
a
2
,
…
,
a
2012
such that
n
2
=
∑
1
≤
i
≤
2012
a
i
p
i
n^2=\sum_{1 \leq i \leq 2012}{{a_i}^{p_i}}
n
2
=
1
≤
i
≤
2012
∑
a
i
p
i
where
p
i
p_i
p
i
is the i-th prime (
p
1
=
2
,
p
2
=
3
,
p
3
=
5
,
p
4
=
7
,
…
p_1=2,p_2=3,p_3=5,p_4=7,\ldots
p
1
=
2
,
p
2
=
3
,
p
3
=
5
,
p
4
=
7
,
…
) and
a
i
>
1
a_i>1
a
i
>
1
for all
i
i
i
?
floor function
number theory unsolved
number theory