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KoMaL A Problems 2021/2022
A. 815
A. 815
Part of
KoMaL A Problems 2021/2022
Problems
(1)
Polynomial equiv factorial modulo prime
Source: KoMaL A815
3/19/2022
Let
q
q
q
be a monic polynomial with integer coefficients. Prove that there exists a constant
C
C
C
depending only on polynomial
q
q
q
such that for an arbitrary prime number
p
p
p
and an arbitrary positive integer
N
≤
p
N \leq p
N
≤
p
the congruence
n
!
≡
q
(
n
)
(
m
o
d
p
)
n! \equiv q(n) \pmod p
n
!
≡
q
(
n
)
(
mod
p
)
has at most
C
N
2
3
CN^\frac {2}{3}
C
N
3
2
solutions among any
N
N
N
consecutive integers.
number theory