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National and Regional Contests
PEN Problems
PEN E Problems
11
11
Part of
PEN E Problems
Problems
(1)
E 11
Source:
5/25/2007
In 1772 Euler discovered the curious fact that
n
2
+
n
+
41
n^2 +n+41
n
2
+
n
+
41
is prime when
n
n
n
is any of
0
,
1
,
2
,
⋯
,
39
0,1,2, \cdots, 39
0
,
1
,
2
,
⋯
,
39
. Show that there exist
40
40
40
consecutive integer values of
n
n
n
for which this polynomial is not prime.
Euler
algebra
polynomial
modular arithmetic