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National and Regional Contests
PEN Problems
PEN Q Problems
12
12
Part of
PEN Q Problems
Problems
(1)
Q 12
Source:
5/25/2007
Prove that if the integers
a
1
a_{1}
a
1
,
a
2
a_{2}
a
2
,
⋯
\cdots
⋯
,
a
n
a_{n}
a
n
are all distinct, then the polynomial
(
x
−
a
1
)
2
(
x
−
a
2
)
2
⋯
(
x
−
a
n
)
2
+
1
(x-a_{1})^{2}(x-a_{2})^{2}\cdots (x-a_{n})^{2}+1
(
x
−
a
1
)
2
(
x
−
a
2
)
2
⋯
(
x
−
a
n
)
2
+
1
cannot be expressed as the product of two nonconstant polynomials with integer coefficients.
algebra
polynomial
Gauss
Polynomials