Advances in Pure Mathematics, 2011, 1, 133-135
doi:10.4236/apm.2011.14026 Published Online July 2011 (http://www.SciRP.org/journal/apm)
Gauss’ Problem, Negative Pell’s Equation and Odd Graphs
Aleksander Grytczuk
Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra,
Zielona Góra, Poland
Department of Mathematics and Applications, Jan Paweł II Western Higher School of Commerce
and International Finance, Zielona Góra, Poland
Received March 30, 2011; revised May 4, 2011; accepted May 15, 2011
Abstract
In this paper we present some results connected with still open problem of Gauss, negative Pell’s equation
and some type graphs. In particular we prove in the Theorem 1 that all real quadratic fields


=
K
Qd,
generated by Fermat’s numbers with 1
2
1
==2
m
m
dF 
1,

2m have not unique factorization. Theorem 2
give a connection of the Gauss problem with primitive Pythagorean triples. Moreover, in final part of our
paper we indicate on some connections of the Gauss problem with odd graphs investigated by Cremona and
Odoni in the papper [1].
Keywords: Fermat Numbers, Class-Number Gauss’ Problem, Odd Graphs
1. Introduction
Let
s
Z
be the set of all square-free positive integers.
Moreover, let denote the class-number of real

hd
quadratic field

=,.
s
K
QddZIt is well-known (Cf.
[2]) that the condition is equivalent to unique
factorization in the ring

=1hd
K
R of the algebraic integers of
the field .
K
The difficult and still open problem posed
by Gauss concern of the existence infinitely many
s
dZ such that It is known that if

hd =1.


=1hd
in the field

=
K
Qd then =, 2, dpq,qr

where
,pq and are primes such that r


34od.qr m
This result has been proved by Hasse in the paper
[3].Another proof of this result has been presented by
Szymiczek [4]. In the paper [5] we give some arithmetic
description of the set
s
Z
and as consequence we also
obtained this result.Many others interesting and impor-
tant results concerning factorization problem have been
given in the papers [1,6-7,9-14]. In this paper we prove
of the following two theorems:
Theorem 1: Let be the
1
2
1
==2 1,
m
m
dF m

2
Fermat’s number and let be the class-
 
1
=m hd hF

number of the real quadratic field

=.
K
QdThen we
have


1mm
MhF
 (*)
where is the Mersenne number. =2 1
m
m
M
Theorem 2: Let
s
dZ

awnd

 here is the
number of all distinct prime divisors of f there is
primitive Pythagorean triple
2,d
d.I
,,

 and positive
relatively prime integers a, b such that
22
=,da b =1ab    (**)
then


>1hd (***)
2. Basic Lemmas
Lemma 1: (H. W. Lu, [13]). Let 2 =4 1
n
dk

where
are positive integers such that ,kn >
1
k and >
1
n.
Then we have


0modhd n

(2.1)
where


hd denote of the class-number of the field


=.
K
Qd
Lemma 2: (A. Grytczuk, F. Luca, M. Wójtowicz,
[15]). The negative Pell equation
22
=1xdy

 (2.2)
has a solution in positive integers x, y if and only if there
is a primitive Pythagorean triple ,,


and positive
relatively prime integers a, b such that
22
=,da b =1ab    (2.3)

134 A. GRYTCZUK
Lemma 3: (A. Grytczuk, J. Grytczuk [5]). Let
s
Z
be
the set of all square-free integers and let p, pj for
be primes. Moreover, let =1,2,...,jk

 
1
=;=...1,1 mod4
sk
AdZdpppk p
 
=;=2,1mod
s
BdZd pp4
2,
k


 
12
=;=2...,
3mod4,= 1, 2,...,
sk
j
CdZd pppk
pjk



 
12
=; =...,3,
3mod4,= 1, 2,...,
sk
j
DdZdpp pk
pj


and


hd be the class-number of the real quadratic
number field

=.
K
Qd
If then ,dABCD


>1.hd
3. Proof of the Theorem 1
For the proof of (*) in the Theorem 1 we use Lemma 1.
First we note that is positive integer, hence we
put and let =2n mom this
fact we obtain
>1k
m
=2k 1>1, soFr2.

22 11
22 2
1
=41=2 21=21=
mm
n
m
dk F


  (3.1)
By (2.1) of Lemma 1 it follows that

nh d (3.2)
Since is m-th Mersenne number
then from (3.1) and (3.2) follows that (*) is true .
=2 1=
m
m
nM

The proof of the Theorem 1 is complete. 
From the Theorem 1 and some property of Fermat
numbers follows the following Corollary:
Corolla ry1: For each positive integer we have 1k
 011 21
... kk FFF hF

 (3.3)
where are Fermat numbers.
2
=2 1
j
j
F
Proof. We use well-known (see, Cf. [16]) the
following identity:
01 1
=... 2;
kk
FFF F

  (3.4) 1k
Since then from (3.4) we obtain
2
=2 1
k
k
F
2k
01 1
21= ...
k
F
FF

 (3.5)
Since thus by (3.5) and (*) it follows
2
2
21=
k
k
M,
the divisibility (3.3) and the proof of Corollary 1 is
finished.

4. Proof of the Theorem 2
Since
s
dZ and


2d   then by the assumption of
the Theorem 2 and Lemma 2 it follows that negative
Pell’s equation (2.2) has a solution in positive relatively
prime integers x, y. From the assumption that


2d  
it follows that there is a prime such that p.pd By
(2.2) it follows that
2
1= 2
x
dy (4.1)
From the well-known properties of divisibility relation
and (4.1) we get
21px  (4.2)
By the relation (4.2) it follows that

21

x
modp (4.3)
From (4.3) we see that 1

is a quadratic residue and
consequently we have
1=1
p



 (4.4)
On the other hand from the property of Legendre’s
symbol (Cf. [17], p. 342) we have

1
2
1=1
p
p




 (4.5)
By (4.4) and (4.5) it follows that the prime number
is the form:
p
=4 1.pk

Since then we see
that

2d  
dA

or dB

and by Lemma 3 it follows that


>1.hd
The proof of the Theorem 2 is complete. 
5. Connections Negative Pell’s Equation with
Gauss’ Problem and Graphs Theory
Let P denote the set of all primes and let




=1<,andthen 14
s
DdZifpPpdpmod

(5.1)
Let n
DD

,


=nd  ; and 1 ,pq P,pqD

.
Then the following relation has been defined by
Cremona and Odoni in the paper [1]:
R
11
,pqRD Dp q

 and


32 3
pxmodq
for some
x
Z

(5.2)
Let n
dD

and 12 =...
n
dpp p

 ; ,
12
<< ... <n pp p
j
pP

, 1jn

.
Now, we form graph

H
d with vertex set


=1,2, ,
n
Sn

 in such way that vertices i, j are
adjacent if and only if ,ppj
ij R for . (5.3)
i
A graph with vertex set n is called as odd
graph when he has of the following property:
G S
Whenever n is partitioned into disjoin union S
X
Y of two non-empty sets ,
X
Y then either exists

A. GRYTCZUK
135
x
X joined in G to an odd number of vertices in
or there exists
Y
y
Y joined in G to an odd number of
vertices in
X
.
Cremona-Odoni Theorem: [1] If n and dD


H
d
,
is an odd graph then is negative Pellian. d
Remark: If the Diophantine equation
has a solution in positive integers
22
xd=1y
x
y
then the number is called as negative Pellian. d
From this Remark, Theorem 2 and The Cremona-
Odoni theorem it follows the following Corollary:
Corollary 2: If and ,
n
dDn2

H
d is an odd
graph, then


>1.hd
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
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