1) The document presents two lemmas showing that when dividing Fibonacci numbers or Lucas numbers by other numbers in their respective sequences, the quotients round to Lucas numbers and the remainders are Fibonacci or Lucas numbers. 2) It provides proofs of the lemmas using known properties of the Fibonacci and Lucas sequences. 3) The document notes that this quotient property does not hold for general recursive sequences defined similarly to the Fibonacci sequence.

1. M. Bicknell-Johnson aN C- P. Spears, Lucas QuolbnlLemmas, Congressus
Numerantium 201, Edited by F, Luca aN P. Stanica, Utilitas Mathemafica Publishing,
Wnnipeg, Canada, 2009, 27927 5.
LUCAS QUOTIENT LEMMAS
MaRronm BrcxuBr.l-JoHNSoN AND CoLrN P,q.ul SpnaRs
In current work pertaining to models of asymmetric cell division and re-
cursive phyllotaxic patterning in biologic structures l3], 14], number sequences
containing F, entries were organized in rectangular tables with F@ columns.
Analysis of data arising from the division of one positive Fibonacci number by
arrother gave a surprising relationship to Lucas numbers: quotienls round off to
Lucas numbers. That the remainders after such divisions are Fibonacci numbers
was known from [1], [2], and [5], but the almost-Lucas quotients in the lemmas
following seem to be new.
Lucas Quotient Lemma 1. When Fo is divided by F-, 3m > p ) m ) 0,
the quotient rounds off (either up or down) to a Lucas number. The remainder
is a Fibonacci number or its negative.
Proof: Vajda [6] Iists the twin equatiorrs (15a) and (15b)
F,+- * (-r)^Fn-* : L^Fn and Fr+- (-l)^Fn-- : F^Ln.
Upon division by F-, (15b) gives
Fn+,,/F^: Ln I (-L)- Fn--fF*'
If Fr--< F-, the quotient is L, and the remainder is f Fr--. If p : I 1
m, the equation above becomes
Fo * : Lp-^ * (-t)^F e-2-fF *
lF
and when p ( 3m so that p 2nr ( m, the fractional part is Iess than one
in absolute value, and the expression rounds off (either up or down) to Lp--.
Note that, when m is even, the remainder is the Fibonacci number Fp-z- and
we round down; if F,p-z- ( 0, we round up, and the quotient obtained with
a calculator is (Lp-- - 1), since
FrlF*: Lp_^ - Fp_z^fF*: (Le_* - 1) + (F^ - F.p_2-)fF*.
In the caiculator case, the positive remainder is
Frr-F,p-2rr: (Frr-z + F--+) * F--o +. . . lFp-z*:L^,s * ...
Equation (15a) yields similar results.
Lucas Quotient Lernma 2. When a Lucas number Lo is divided by L^,
3m ) p > m > 0, the quotient rounds off to a Lucas number. The (non-zero)
remainder is either a Lucas number or its negative.
Proof: Apply Equation (17a) from 16] and analyze as in Lemma 1:
Ln+^ * (-I)^L- *: L*L,.

2. M. Bicknell-Johnson and C. F. Spears, Lucas QuotientLemmas, Congressus
Numerantium 201, Edited by F- Luca and P. Stanica, Utititas Mathematica Publishing,
VWnnipeg, Canada, 2AA9, 27 3-27 5.
From [5], the Fibonacci and Lucas sequences are the only Fibonacci-like
sequences possessing the property that division of a member of the sequence
by a (non-zero) member of that same sequence yields least positive or negative
residues that are either zero or a mernber of the original sequence.
For the Fibonacci-like sequence deflned by G.+r : G", * G",-r, G0, Gr ar*
bitrary positive integers, neither the Lucas quotient property nor the rernainder
property holds in general. For example, for the sequence arising frorn G6 : /,
Gr : 3, {..., 26, -15, 11, -4,7, 3, 10, 13, 23, 36,59, ...}, division of 59 by
10 gives 5 remainder 9 or 6 remainder (-1); 9 and f1) do not appear in the
sequence. While the sequences {G",} have t}re property t}rat {G,} is congru-
ent to a sequence made of the original sequence and negatives of those values,
G, = t G. (mod G6), those subsequences are actuaily remainders of the
divisor when G./Ga for onll the Fibonacci and Lucas sequences [5].
At first glance, Eq. (f0a) from [6] seerns to apply:
G",+., * (-)"' Gn-^: L^Gn.
If rn is odd and G.,- < G., there are some cases of Lucas quotients
paired with remainders witliin the sequence {G.}. However, {G,} Iacks the
symrnetry about Go of the Fiboriacci sequence.
RBpBRBNcnS
1. U. Alfred fBrousseau] , Erpl,oring Fibonacci Residues, The Fiborracci Quar-
terly 2 (1964),42.
2. J.H. Halton, On F'ibonacci Res'id,ues, The Fibonacci Quarterly 2 (1964),
217-218.
3. C. P. Spears and M. Bickneli-Johnson. "Asymmetric Cell Divisiorr: Birro-
nrial Identities for Age Analysis of Mortal vs. Irnmortal Ttees." Applicati,ons of
Fibortacci. Nurnbers. Volume 7: Edited by G. E. Bergum et ai. Kluwer Academic
Publishers, Dordrecht, 1998: 377-391.
4. C. P. Spears, M. Bicknell-Johnson, and J. J. Yan, F'ibonacci Phyllotani.s
bg Asynr,rnetric Cell Diu'isiort, Thirteerrth Internatiorral Conference on Fibonacci
Numbers and Their Applicatiorrs, July 7-11, 2008, Patras, Greece.
5. L. Taylor, Residues of F'ibonacci-I'ike Sequences, The Fibonacci Quarterly
5 (1967),298-304.
6. S. Vajda, Fi,bonacci, and Lucas Numbers and the Golden Section, Wlley
and Sons, New York, 1989.
AMS Classification numbers: 11865, 11839
665 FernleuE AvENUE, SaNT.q. Cr,aR.a., CA 95051
E-mail addres s : marjohnson@mac.com
4851 Oax Vrsra DRrvE, CARMrcHarr,, CA 95608
E-mai,l add,ress : cpspears@aol.com