Problem (2) Find a composite or which divides both (see Fermat pseudoprime) and the Fibonacci number (see Lucas pseudoprime), or prove that there is no such .
For every finite multiplicative group , let () denote the smallest integer so that every sequence of elements of has a subsequence of length (length ) which has product equal to 1 in some order.
Conjecture Let and . Then for any neighborhood there is such that is periodic point of
There is an analogous conjecture for flows ( vector fields . In the case of diffeos this was proved by Charles Pugh for . In the case of Flows this has been solved by Sushei Hayahshy for . But in the two cases the problem is wide open for
Let denote the set of all permutations of . Let and denote respectively the number of increasing and the number of decreasing sequences of contiguous numbers in . Let denote the set of subsequences of with length at least three. Let denote .
A permutation is called a Roller Coaster permutation if . Let be the set of all Roller Coaster permutations in .
Conjecture For ,
\item If , then . \item If , then with .
Conjecture (Odd Sum conjecture) Given ,
\item If , then is odd for . \item If , then for all .
Conjecture Let and are monovalued, entirely defined funcoids with . Then there exists a pointfree funcoid such that (for every filter on ) (The join operation is taken on the lattice of filters with reversed order.)
A positive solution of this problem may open a way to prove that some funcoids-related categories are cartesian closed.
Problem Consider the set of all topologically inequivalent polyhedra with edges. Define a form parameter for a polyhedron as where is the number of vertices. What is the distribution of for ?
The star chromatic index of a graph is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi-colored.
Question Is it true that for every (sub)cubic graph , we have ?
Conjecture Let be a simple -uniform hypergraph, and assume that every set of points is contained in at most edges. Then there exists an -edge-coloring so that any two edges which share vertices have distinct colors.
Problem The valency-variety of a graph is the number of different degrees in . Is the chromatic number of any graph with at least two vertices greater than
Conjecture Is the approximation ratio for the Maximum Edge Disjoint Paths (MaxEDP) or the Maximum Integer Multiflow problem (MaxIMF) bounded by a constant in -outerplanar graphs or tree-width graphs?
Conjecture Let if is odd and if is even. Let . Assume we start with some number and repeatedly take the of the current number. Prove that no matter what the initial number is we eventually reach .
Conjecture If every second positive integer except 2 is remaining, then every third remaining integer except 3, then every fourth remaining integer etc. , an infinite number of the remaining integers are prime.
An oriented colouring of an oriented graph is assignment of colours to the vertices such that no two arcs receive ordered pairs of colours and . It is equivalent to a homomorphism of the digraph onto some tournament of order .
A covering design, or covering, is a family of -subsets, called blocks, chosen from a -set, such that each -subset is contained in at least one of the blocks. The number of blocks is the covering’s size, and the minimum size of such a covering is denoted by .
Problem Find a closed form, recurrence, or better bounds for . Find a procedure for constructing minimal coverings.
Conjecture Let be a circuit in a bridgeless cubic graph . Then there is a five cycle double cover of such that is a subgraph of one of these five cycles.
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or 
which divides both 
(see 
(see 
. 
, let 
(
) denote the smallest integer 
so that every sequence of 
(length 
) which has product equal to 1 in some order. 
for every finite group 
of graphs is 
-bounded if there exists a function 
so that every 
satisfies 
. 
, the family of graphs with no induced subgraph isomorphic to 
and 
. Then for any neighborhood 
there is 
such that 
is periodic point of 
vector fields . In the case of diffeos this was proved by Charles Pugh for 
. In the case of Flows this has been solved by Sushei Hayahshy for 
of 
denote the set of all permutations of ![$ [n]=\set{1,2,\ldots,n} $](/files/tex/470bd09059264370b76b4da90b77dc370c7e0e7c.png)
. Let 
and 
denote respectively the number of increasing and the number of decreasing sequences of contiguous numbers in 
. Let 
denote the set of subsequences of 
denote 
.
is called a Roller Coaster permutation if 
. Let 
be the set of all Roller Coaster permutations in 
,
, then 
. \item If 
, then 
with 
. 
,
is odd for 
. \item If 
for all 
and 
are monovalued, entirely defined funcoids with 
. Then there exists a pointfree funcoid 
such that (for every filter 
on 
) 
(The join operation is taken on the lattice of filters with reversed order.) 
edges. Define a form parameter for a polyhedron as 
where 
is the number of vertices. What is the distribution of 
for 
? 
has equivalence covering number 
. 
be a vector space of dimension 
be bases. Then there exist 
each of which is a base. 
to be the maximum degree, 
to be the size of the largest 
to be the chromatic number of 
for every graph 
.
, the union of any 
-degenerate graph. 
for principal funcoid 
and a set 
of funcoids of appropriate sources and destinations. 
of a graph 
? 
be integers. Set 
and 
for 
. Eventually we have 
; put 
.
, since 
, 
, 
, 
, 
, 
, 
, 
.
.
, there exist positive integers 
, 
such that 
.
be a simple 
-
points is contained in at most 
edges. Then there exists an 
-edge-coloring so that any two edges which share 
contains 
of a graph 
denote ''
divides 
''. The mimic function in number theory is defined as follows [1].
divisible by 
, the mimic function, 
, is given by,
. 
if 
if 
. Assume we start with some number 
. 
into 
so that the graph induced by 
is acyclic for 
. 
of colours to the vertices such that no two arcs receive ordered pairs of colours 
and 
. It is equivalent to a homomorphism of the digraph onto some tournament of order 
. In other words, simplify this formula. 
is at most 
.
, 
so that 
does not contain an odd edge-cut. 
, there is only a finite number of good-edge-labeling critical graphs with average degree less than 
be positive semidefinite, by Jensen's inequality, it is easy to see ![$ [tr(A^s+B^s)]^{\frac{1}{s}}\leq [tr(A^r+B^r)]^{\frac{1}{r}} $](/files/tex/f5fa90ccf8c1dde9b9bf1f21399b3b5428dab81d.png)
, whenever 
.
, is it still valid?
the following inequality holds 
people admits a solution is 
. 
paths. 
? 
covering design, or covering, is a family of 
-subset is contained in at least one of the blocks. The number of blocks is the covering’s size, and the minimum size of such a covering is denoted by 
.
. Find a procedure for constructing minimal coverings. 
-minor. 
has distinct subset sums if distinct subsets of 
whenever 
has distinct subset sums. 
? 
Square free? 
if 
if 
; \item 
; \item 
. 
be an elliptic curve over a number field 
. Then the order of the zeros of its 
-function, 
, at 
is the Mordell-Weil rank of 
. 
points in the plane in general position contains a subset of 
be a circuit in a bridgeless cubic graph 
-connected cubic planar graph can be decomposed into two Hamilton cycles. 
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