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Danube Competition in Mathematics
2012 Danube Mathematical Competition
3
Danube 2012 #3
Danube 2012 #3
Source:
December 28, 2015
number theory
Problem Statement
Let
p
p
p
and
q
,
p
<
q
,
q, p < q,
q
,
p
<
q
,
be two primes such that
1
+
p
+
p
2
+
.
.
.
+
p
m
1 + p + p^2+...+p^m
1
+
p
+
p
2
+
...
+
p
m
is a power of
q
q
q
for some positive integer
m
m
m
, and
1
+
q
+
q
2
+
.
.
.
+
q
n
1 + q + q^2+...+q^n
1
+
q
+
q
2
+
...
+
q
n
is a power of
p
p
p
for some positive integer
n
n
n
. Show that
p
=
2
p = 2
p
=
2
and
q
=
2
t
ā
1
q = 2^t-1
q
=
2
t
ā
1
where
t
t
t
is prime.
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