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Putnam
1983 Putnam
A3
A3
Part of
1983 Putnam
Problems
(1)
1+2n+3n^2+...+(p-1)n^(p-2), F(a)!=F(b) mod p
Source: Putnam 1983 A3
9/14/2021
Let
p
p
p
be an odd prime and let
F
(
n
)
=
1
+
2
n
+
3
n
2
+
…
+
(
p
−
1
)
n
p
−
2
.
F(n)=1+2n+3n^2+\ldots+(p-1)n^{p-2}.
F
(
n
)
=
1
+
2
n
+
3
n
2
+
…
+
(
p
−
1
)
n
p
−
2
.
Prove that if
a
a
a
and
b
b
b
are distinct integers in
{
0
,
1
,
2
,
…
,
p
−
1
}
\{0,1,2,\ldots,p-1\}
{
0
,
1
,
2
,
…
,
p
−
1
}
then
F
(
a
)
F(a)
F
(
a
)
and
F
(
b
)
F(b)
F
(
b
)
are not congruent modulo
p
p
p
.
number theory