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M2605
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(1)
Strange NT
Source: Romanian Masters in Mathematics 2020, Problem 6
3/1/2020
For each integer
n
≥
2
n \geq 2
n
≥
2
, let
F
(
n
)
F(n)
F
(
n
)
denote the greatest prime factor of
n
n
n
. A strange pair is a pair of distinct primes
p
p
p
and
q
q
q
such that there is no integer
n
≥
2
n \geq 2
n
≥
2
for which
F
(
n
)
F
(
n
+
1
)
=
p
q
F(n)F(n+1)=pq
F
(
n
)
F
(
n
+
1
)
=
pq
.Prove that there exist infinitely many strange pairs.
RMM
RMM 2020
number theory