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Putnam
1985 Putnam
B2
B2
Part of
1985 Putnam
Problems
(1)
Putnam 1985 B2
Source:
8/5/2019
Define polynomials
f
n
(
x
)
f_{n}(x)
f
n
(
x
)
for
n
≥
0
n \geq 0
n
≥
0
by
f
0
(
x
)
=
1
,
f
n
(
0
)
=
0
f_{0}(x)=1, f_{n}(0)=0
f
0
(
x
)
=
1
,
f
n
(
0
)
=
0
for
n
≥
1
,
n \geq 1,
n
≥
1
,
and
d
d
x
f
n
+
1
(
x
)
=
(
n
+
1
)
f
n
(
x
+
1
)
\frac{d}{d x} f_{n+1}(x)=(n+1) f_{n}(x+1)
d
x
d
f
n
+
1
(
x
)
=
(
n
+
1
)
f
n
(
x
+
1
)
for
n
≥
0.
n \geq 0 .
n
≥
0.
Find, with proof, the explicit factorization of
f
100
(
1
)
f_{100}(1)
f
100
(
1
)
into powers of distinct primes.
Putnam