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International Contests
Baltic Way
1996 Baltic Way
10
10
Part of
1996 Baltic Way
Problems
(1)
d(a^n-1) is at least n if a^n+1 is prime
Source: Baltic Way 1996 Q10
3/19/2011
Denote by
d
(
n
)
d(n)
d
(
n
)
the number of distinct positive divisors of a positive integer
n
n
n
(including
1
1
1
and
n
n
n
). Let
a
>
1
a>1
a
>
1
and
n
>
0
n>0
n
>
0
be integers such that
a
n
+
1
a^n+1
a
n
+
1
is a prime. Prove that
d
(
a
n
−
1
)
≥
n
d(a^n-1)\ge n
d
(
a
n
−
1
)
≥
n
.
number theory
relatively prime
number theory proposed