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Number Theory problem on sequence
Number Theory problem on sequence
Source: Romania 2017 IMO TST 4, problem 1
March 18, 2018
number theory
Problem Statement
Let m be a positive interger, let
p
p
p
be a prime, let
a
1
=
8
p
m
a_1=8p^m
a
1
=
8
p
m
, and let
a
n
=
(
n
+
1
)
a
n
−
1
n
a_n=(n+1)^{\frac{a_{n-1}}{n}}
a
n
=
(
n
+
1
)
n
a
n
−
1
,
n
=
2
,
3...
n=2,3...
n
=
2
,
3...
. Determine the primes
p
p
p
for which the products
a
n
(
1
−
1
a
1
)
(
1
−
1
a
2
)
.
.
.
(
1
−
1
a
n
)
a_n(1-\frac{1}{a_1})(1-\frac{1}{a_2})...(1-\frac{1}{a_n})
a
n
(
1
−
a
1
1
)
(
1
−
a
2
1
)
...
(
1
−
a
n
1
)
,
n
=
1
,
2
,
3...
n=1,2,3...
n
=
1
,
2
,
3...
are all integral.
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