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Romanian Masters of Mathematics Collection
2018 Romanian Master of Mathematics Shortlist
N2
N2
Part of
2018 Romanian Master of Mathematics Shortlist
Problems
(1)
b-ary Fibonacci Numbers
Source: 2018 RMM Shortlist N2
2/21/2019
Prove that for each positive integer
k
k
k
there exists a number base
b
b
b
along with
k
k
k
triples of Fibonacci numbers
(
F
u
,
F
v
,
F
w
)
(F_u,F_v,F_w)
(
F
u
,
F
v
,
F
w
)
such that when they are written in base
b
b
b
, their concatenation is also a Fibonacci number written in base
b
b
b
. (Fibonacci numbers are defined by
F
1
=
F
2
=
1
F_1 = F_2 = 1
F
1
=
F
2
=
1
and
F
n
+
2
=
F
n
+
1
+
F
n
F_{n+2} = F_{n+1} + F_n
F
n
+
2
=
F
n
+
1
+
F
n
for all positive integers
n
n
n
.) Proposed by Serbia
number theory
Fibonacci