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2017 Brazil National Olympiad
6.
6.
Part of
2017 Brazil National Olympiad
Problems
(1)
Primes that divide a³-3a+1
Source: Question 6 - Brazilian Mathematical Olympiad 2017
12/7/2017
6. Let
a
a
a
be a positive integer and
p
p
p
a prime divisor of
a
3
−
3
a
+
1
a^3-3a+1
a
3
−
3
a
+
1
, with
p
≠
3
p \neq 3
p
=
3
. Prove that
p
p
p
is of the form
9
k
+
1
9k+1
9
k
+
1
or
9
k
−
1
9k-1
9
k
−
1
, where
k
k
k
is integer.
number theory
Divisibility
prime
Brazilian Math Olympiad
Brazilian Math Olympiad 2017
group theory