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Miklós Schweitzer
2019 Miklós Schweitzer
3
3
Part of
2019 Miklós Schweitzer
Problems
(1)
All four gcds are at least c*sqrt(n)
Source: Miklós Schweitzer 2019, Problem 3
12/27/2019
Prove that there are infinitely many integers
m
m
m
,
n
n
n
, such that
1
<
m
<
n
1 < m < n
1
<
m
<
n
, and the greatest common divisors
(
m
,
n
)
(m, n)
(
m
,
n
)
,
(
m
,
n
+
1
)
(m, n+1)
(
m
,
n
+
1
)
,
(
m
+
1
,
n
)
(m+1, n)
(
m
+
1
,
n
)
and
(
m
+
1
,
n
+
1
)
(m+1, n+1)
(
m
+
1
,
n
+
1
)
are all greater than
n
/
999
\sqrt{n}/999
n
/999
.
number theory