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National and Regional Contests
PEN Problems
PEN E Problems
36
36
Part of
PEN E Problems
Problems
(1)
E 36
Source:
5/25/2007
Prove that there are infinitely many twin primes if and only if there are infinitely many integers that cannot be written in any of the following forms:
6
u
v
+
u
+
v
,
6
u
v
+
u
−
v
,
6
u
v
−
u
+
v
,
6
u
v
−
u
−
v
,
6uv+u+v, \;\; 6uv+u-v, \;\; 6uv-u+v, \;\; 6uv-u-v,
6
uv
+
u
+
v
,
6
uv
+
u
−
v
,
6
uv
−
u
+
v
,
6
uv
−
u
−
v
,
for some positive integers
u
u
u
and
v
v
v
.