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2016 IMO Shortlist
N4
n^k + mn^l + 1 divides n^(k+1) - 1
n^k + mn^l + 1 divides n^(k+1) - 1
Source: 2016 IMO Shortlist N4
July 19, 2017
number theory
IMO Shortlist
Problem Statement
Let
n
,
m
,
k
n, m, k
n
,
m
,
k
and
l
l
l
be positive integers with
n
≠
1
n \neq 1
n
=
1
such that
n
k
+
m
n
l
+
1
n^k + mn^l + 1
n
k
+
m
n
l
+
1
divides
n
k
+
l
−
1
n^{k+l} - 1
n
k
+
l
−
1
. Prove that[*]
m
=
1
m = 1
m
=
1
and
l
=
2
k
l = 2k
l
=
2
k
; or [*]
l
∣
k
l|k
l
∣
k
and
m
=
n
k
−
l
−
1
n
l
−
1
m = \frac{n^{k-l}-1}{n^l-1}
m
=
n
l
−
1
n
k
−
l
−
1
.
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