OPEN PROBLEMS from the FLAGSTAFF
CONFERENCE June 2002
Proposer 
Brief Statement of Problem 
Further notes 
Curtis
Cooper & Michael Wiemann 
Fix m: Define x_{1 }= 0, x_{n }= x^{m } x_{n
 1} Let P_{m}(n) be a
solution to the above Find a closed formula for P_{m}(n) 
*2
Known formulations of P_{m}(n) 
Glenn
Hurlbert & Rob Hochberg 
Inscribe ngon in
semicircle Require integer sides If n=3==>Pythagoreean
triples If n=4=>Pythagoreean
Quadruple Are there infinitely many n with nested Pythagoreean ntuples? 

Eric
Egge & Toufik Mansour 
Fix n. Let S_{n} be
all npermutations Let R be a set of
Permutations Let S_{n}(R)=(s in
S_{n}: s avoids R) Find formula for S_{n}(R) Find S_{n}(R)
when R [avoids/contains] 123 and [contains/avoids] r>1
other patterns. 
*5.
Known results on S_{n}(R) 
Arthur Benjamin & Jennifer
Quinn 
Lemma: F_{n} and L_{n} count the number
of dominosquare tilings of n x 1 boards/bracelets. Using this lemma find proofs for all formula in Vajda 
*6.
Precise statement of
lemma
*7.
What is known / references 
Heiko
Harborth 
Take an empty Pascals
triangle Fill the 2 diagonal borders
with finite # 1,0 Fill rest of triangle by
addition modulo 2 Is every n equal to
the number of 1s in triangle for some initial borders? Given n, Find minimal borders such that # 1 in Pascal triangle = n? 

Clark
Kimberling 
1, 1, 2, 3, 5, 8, 4, 12,
6,..... Start sequence with 1,1 If S_{n} /2 not in list then S_{n+1}=S_{n}
/ 2; else S_{n+1} = S_{n}+S_{n1} Is every natural number in this list? 

Heiko
Harborth 
Pick a base b to represent numbers in Let S_{1 }= a be arbitrary. Let S_{n+1} =
S_{n} + reversal of S_{n} Find a and b such that no palindrom occurs in the sequence S_{n} 

Heiko
Harborth 
Develop algorithms for magic ngons (Generalize magic square theory) 
*10.
Some elementary facts & examples 
Daniel
Fielder 
A Deception problem 
*11.
See footnote for
full statement 
Russell
Hendel 
Given k, Solve F_{n} = P_{k,m} for (n,m) Solve generalizations 
*13.
A long list of references 
Larry
Somer 
Let
u_{n+2} = a u_{n+1} + b u_{n} ; Let
u_{0}=2 and u_{1 } be
arbitrary (For
u_{1}=1, a=b=1, we obtain the Lucas numbers) Discuss the density of u_{n}
with prime factors congruent to 1 modulo 4 

 
*1 See section 8 of Divisibility
of an FL Type Convolution, section 8 by the authors
RETURN
P_{0}(n)=1 iff(n is odd,1,0),
P_{1}(n)=n/2 + if(n odd,1,0)/2, P_{2}(n)=n^{2}/2 +n/2 P_{m}(n) = 0.5 *( (n+1)^{m}
S C(m,i) P_{mi }(n)) Where C(m,i) is the binomial
coefficient, m chose 1, and iff(Condition,value1,value2)
equals value1 if condition
is true and equals value2 otherwise
RETURN
*3
Let S_{n}
denote the set of all permutations of {1,...,n} written in oneline notation.
Suppose p_{1}, p_{2} in S_{n}. We say p_{1} avoids p_{2} if
p_{1} contains no subsequence with all the same pairwise comparisons of
p_{2}.
*4.
Pattern avoidance
has applicability to the areas of Singularities in Shubert varieties, Chebychev
Polynomials of the 2nd kind, rook polynomials for a rectangular board and
various sorting algorithms
RETURN
*5.
Egge and Mansour
in their paper 132avoiding Twostack Sortable Permutations, Fibonacci
Numbers and Pell Numbers, found explicit formula for S_{n}(R)
(Where   denotes cardinality) when R [avoidscontains] 123 but
[contains\avoids] r=1 other patterns. The proofs however involved generating
functions.
*6.
Define a domino
and square respectively as a 2 x 1 and 1 x 1 board. Then F_{n} is the
number of distinct dominosquare tilings of an n x 1 board and L_{n} is
the number of distinct tilings of an n x 1 bracelet (Where a bracelet is an n x
1 board whose first and last squares are adjacent).
RETURN
*7.
Roughly 88% of
the formulae in Vajda have been proved combinatorically. See the
followin references Benjamin A.T. and Quinn J.J. Recounting Fibonacci and Lucas
Identities, College Math Journal 30.5, 1999 Benjamin A.T. and Quinn
J.J. Random approaches to Fibonacci Identities, Amer Math
Mon;107.6,2000
Vajda, S Fibonacci
& Lucas Numbers & the Golden Section: Theory and Applications. Wiley
& Sons, NY 1989
RETURN
*8.
The following
similar problem appeared in Crux Mathematicorum Consider the
following sequence 1,3,9,4,2,5,18.... Let a(n) = Int(a(n1) / 2) if this member is
not already there Let a(n) = 3a(n1) otherwise Show that every
natural number occurs in the sequence a(n)
RETURN
*9.
There is no
solution for base 10. Solutions are known for bases that are powers of 2.
RETURN
*10
The following
magic hexagon was presented
Some elementary
necessary conditions for magic ngons are the following If an ngon, with r
rows, is filled with 1,...,n then each row
must sum to
*11.
Here is the full
statement of the problem  Throughout
let n vary over the set (1,2,3)  Let p be a prime  Let f_{n}(x)
be irreducible polynomials modulo p  Let r_{n}
be positive integers  Let t_{n}
be the smallest solutions of the simultaneous eq. 1  x^{ tn} = 0 (mod f_{n}(x) ^{rn}) Show that the above
implies 1 
x ^{LCM(tn)} = 0 mod (Product f_{n}(x) ^{rn})
RETURN
*12.
This problem was
motivated by one of the opening remarks of the Dean of Northern Arizona
University that “ This is the 10th Fibonacci conference 1+2+...+10=55 F_{10} = 55" An obvious generalization is Find all pairs (n,m) such that F_{n}
= T_{m}, where Tm is the mth triangular number More generally if G_{n} and H_{m} are arbitrary
sequences defined by a recursive equation find all (n,m) such that G_{n} = H_{m.}
RETURN
*13.
Special cases of the
problem have been solved. Florian Luca was kind enough to supply a Bibliography
of papers Ming, L. On triangular Fibonacci numbers,
Fibo. Quart. 27 (1989), 98 Ming, L. On
triangular Lucas numbers, Applications of Fibonacci Numbers vol. 4, Editors,
Bergum, Horadam, Philippou, Kluwer Acad. Publ. Dordrech, the Netherlands,
1991, 98 Ming, L. Pentagonal
numbers in the Fibonacci sequence, Appl. of Fibonacci numbers,
vol. 6, Kluwer, Dordrecht, the Netherlands, 1994, 349354. Ming L., Pentagonal numbers in the Lucas
sequence, Portugaliae Math. 53 (1996), 352329. What these papers
have in common is that the proof if Jacobi symbol based and they provide
succesive refinements of an idea originated in J.H.E. Cohn, On square Fibonacci numbers, J.
London Math. Soc. 39 (1964), 537540. Other papers are Wayne McDaniel, Pronic Fibonacci and Lucas
numbers, Fibo. Quart. 1998, 5659 & 6062; Luca, Florian, Appl.
of Fibonacci numbers, vol. 8, Kluwer, Dordrecht, the
Netherlands, 1999, 241249). General finiteness
results for F_n=P(x) or L_n=P(x) with P a polynomial of degree >1 appear
in Nemes, Petho:
Polynomial values in Linear Recurrences, II, Journal of
Number Theory 24 (1986), 4753 I found the
following relevant problem B875 FQ
37.2 1999; 3 is the only solution to T_{n} = a Fermat Number
RETURN
*14.
Somer proved
that If b is a square and a has prime factors congruent to 1 modulo 4 then all u_{n, }n>2, have prime
factors congruent to 1 modulo 4. Divisibility of the Lucas numbers by prime
factors congruent to 1 modulo 4 has
been studied
extensively and there are many known numerical results. The above
problem is an attempt to generalize. RETURN

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Hendel.;
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