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8
M 8
M 8
Source:
May 25, 2007
algebra
polynomial
Recursive Sequences
Problem Statement
The Bernoulli sequence
{
B
n
}
n
≥
0
\{B_{n}\}_{n \ge 0}
{
B
n
}
n
≥
0
is defined by
B
0
=
1
,
B
n
=
−
1
n
+
1
∑
k
=
0
n
(
n
+
1
k
)
B
k
(
n
≥
1
)
B_{0}=1, \; B_{n}=-\frac{1}{n+1}\sum^{n}_{k=0}{{n+1}\choose k}B_{k}\;\; (n \ge 1)
B
0
=
1
,
B
n
=
−
n
+
1
1
k
=
0
∑
n
(
k
n
+
1
)
B
k
(
n
≥
1
)
Show that for all
n
∈
N
n \in \mathbb{N}
n
∈
N
,
(
−
1
)
n
B
n
−
∑
1
p
,
(-1)^{n}B_{n}-\sum \frac{1}{p},
(
−
1
)
n
B
n
−
∑
p
1
,
is an integer where the summation is done over all primes
p
p
p
such that
p
∣
2
k
−
1
p| 2k-1
p
∣2
k
−
1
.
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