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Turkey MO (2nd round)
2011 Turkey MO (2nd round)
4
Turkey NMO 2011 Problem 4
Turkey NMO 2011 Problem 4
Source:
December 10, 2011
LaTeX
modular arithmetic
number theory unsolved
number theory
legendre s theorem
Problem Statement
a
1
=
5
a_{1}=5
a
1
=
5
and
a
n
+
1
=
a
n
3
−
2
a
n
2
+
2
a_{n+1}=a_{n}^{3}-2a_{n}^{2}+2
a
n
+
1
=
a
n
3
−
2
a
n
2
+
2
for all
n
≥
1
n\geq1
n
≥
1
.
p
p
p
is a prime such that
p
=
3
(
m
o
d
4
)
p=3(mod 4)
p
=
3
(
m
o
d
4
)
and
p
∣
a
2011
+
1
p|a_{2011}+1
p
∣
a
2011
+
1
. Show that
p
=
3
p=3
p
=
3
.
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