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National and Regional Contests
PEN Problems
PEN E Problems
31
31
Part of
PEN E Problems
Problems
(1)
E 31
Source:
1/20/2013
Suppose
n
n
n
and
r
r
r
are nonnegative integers such that no number of the form
n
2
+
r
−
k
(
k
+
1
)
(
k
∈
N
)
n^2+r-k(k+1) \text{ }(k\in\mathbb{N})
n
2
+
r
−
k
(
k
+
1
)
(
k
∈
N
)
equals to
−
1
-1
−
1
or a positive composite number. Show that
4
n
2
+
4
r
+
1
4n^2+4r+1
4
n
2
+
4
r
+
1
is
1
1
1
,
9
9
9
, or prime.
pen
number theory