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Bulgaria National Olympiad
1974 Bulgaria National Olympiad
Problem 2
Problem 2
Part of
1974 Bulgaria National Olympiad
Problems
(1)
P(m^n)+Q([0,k]) is composite for infinitely many k
Source: Bulgaria 1974 P2
6/20/2021
Let
f
(
x
)
f(x)
f
(
x
)
and
g
(
x
)
g(x)
g
(
x
)
be non-constant polynomials with integer positive coefficients,
m
m
m
and
n
n
n
are given natural numbers. Prove that there exists infinitely many natural numbers
k
k
k
for which the numbers
f
(
m
n
)
+
g
(
0
)
,
f
(
m
n
)
+
g
(
1
)
,
…
,
f
(
m
n
)
+
g
(
k
)
f(m^n)+g(0),f(m^n)+g(1),\ldots,f(m^n)+g(k)
f
(
m
n
)
+
g
(
0
)
,
f
(
m
n
)
+
g
(
1
)
,
…
,
f
(
m
n
)
+
g
(
k
)
are composite.I. Tonov
Polynomials
number theory