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Czech-Polish-Slovak Match
2018 Czech-Polish-Slovak Match
6
6
Part of
2018 Czech-Polish-Slovak Match
Problems
(1)
Fantastic integers n = a+1/a+b+1/b [Czech-Polish-Slovak Match 2018]
Source: Czech-Polish-Slovak Match 2018, Problem 6
7/2/2018
We say that a positive integer
n
n
n
is fantastic if there exist positive rational numbers
a
a
a
and
b
b
b
such that
n
=
a
+
1
a
+
b
+
1
b
.
n = a + \frac 1a + b + \frac 1b.
n
=
a
+
a
1
+
b
+
b
1
.
(a) Prove that there exist infinitely many prime numbers
p
p
p
such that no multiple of
p
p
p
is fantastic. (b) Prove that there exist infinitely many prime numbers
p
p
p
such that some multiple of
p
p
p
is fantastic.Proposed by Walther Janous, Austria
number theory