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Baltic Way
1999 Baltic Way
18
18
Part of
1999 Baltic Way
Problems
(1)
At most one factorization of m into the integers ab
Source: Baltic Way 1999
12/23/2010
Let
m
m
m
be a positive integer such that
m
=
2
(
m
o
d
4
)
m=2\pmod{4}
m
=
2
(
mod
4
)
. Show that there exists at most one factorization
m
=
a
b
m=ab
m
=
ab
where
a
a
a
and
b
b
b
are positive integers satisfying
0
<
a
−
b
<
5
+
4
4
m
+
1
0<a-b<\sqrt{5+4\sqrt{4m+1}}
0
<
a
−
b
<
5
+
4
4
m
+
1
modular arithmetic
number theory proposed
number theory