Gauss’ Problem, Negative Pell’s Equation and Odd Graphs

doi:10.4236/apm.2011.14026

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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

E-mail: [email protected]

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





Qd,

generated by Fermat’s numbers with 1

==2

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

be the set of all square-free positive integers.

Moreover, let denote the class-number of real



quadratic field



=,.

QddZIt is well-known (Cf.

[2]) that the condition is equivalent to unique

factorization in the ring



=1hd

R of the algebraic integers of

the field .

The difficult and still open problem posed

by Gauss concern of the existence infinitely many

dZ such that It is known that if



hd =1.





=1hd

in the field



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

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

==2 1,

dF m



2

Fermat’s number and let be the class-



hd hF



number of the real quadratic field



QdThen we

have





1mm

MhF

 (*)

where is the Mersenne number. =2 1

M

Theorem 2: Let



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

=,da b =1ab



  (**)

then





>1hd (***)

2. Basic Lemmas

Lemma 1: (H. W. Lu, [13]). Let 2

=4 1



where

are positive integers such that

,kn >

k and >

Then we have





0modhd n



(2.1)

where





hd denote of the class-number of the field





Lemma 2: (A. Grytczuk, F. Luca, M. Wójtowicz,

[15]). The negative Pell equation

=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

=,da b =1ab



  (2.3)

134 A. GRYTCZUK

Lemma 3: (A. Grytczuk, J. Grytczuk [5]). Let

the set of all square-free integers and let p, pj for

be primes. Moreover, let

=1,2,...,jk







=;=...1,1 mod4

AdZdpppk p





=;=2,1mod

BdZd pp4

2,









=;=2...,

3mod4,= 1, 2,...,

CdZd pppk

pjk











=; =...,3,

3mod4,= 1, 2,...,

DdZdpp pk





and





hd be the class-number of the real quadratic

number field



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

=2k 1>1, soFr2.



22 11

22 2

=41=2 21=21=

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=

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 1

F

Proof. We use well-known (see, Cf. [16]) the

following identity:

01 1

=... 2;

FFF F



  (3.4) 1k

Since then from (3.4) we obtain

=2 1

F

01 1

21= ...



 (3.5)

Since thus by (3.5) and (*) it follows

21=

M,

the divisibility (3.3) and the proof of Corollary 1 is

finished.



4. Proof of the Theorem 2

Since

dZ and







 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









it follows that there is a prime such that

p.pd By

(2.2) it follows that

1= 2

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





modp (4.3)

From (4.3) we see that 1



is a quadratic residue and

consequently we have

1=1







 (4.4)

On the other hand from the property of Legendre’s

symbol (Cf. [17], p. 342) we have



1=1









 (4.5)

By (4.4) and (4.5) it follows that the prime number

is the form:

=4 1.pk



Since then we see

that









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

DdZifpPpdpmod



(5.1)

Let n



,





=nd



; and 1

,pq P,pqD



Then the following relation has been defined by

Cremona and Odoni in the paper [1]:

,pqRD Dp q



 and





32 3

pxmodq

for some



(5.2)

Let n



and 12

=...

dpp p



 ; ,

<< ... <n

pp p



, 1jn



.

Now, we form graph



d with vertex set





=1,2, ,



 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

Y of two non-empty sets ,

Y then either exists

A. GRYTCZUK

135

X joined in G to an odd number of vertices in

or there exists

Y joined in G to an odd number of

vertices in

Cremona-Odoni Theorem: [1] If n and

dD





is an odd graph then is negative Pellian.

Remark: If the Diophantine equation

has a solution in positive integers

xd=1y

then the number is called as negative Pellian.

From this Remark, Theorem 2 and The Cremona-

Odoni theorem it follows the following Corollary:

Corollary 2: If and ,

dDn2



d is an odd

graph, then





>1.hd

6. References

[1] J. E. Cremona and R. W. K. Odoni, “Some Density Re-

sults for Negative Pell Equations: An Application on

Graphs Theory,” Journal of the London Mathematical

Society, Vol. 39, No. 1, 1993, pp. 16-28.

doi:10.1112/jlms/s2-39.1.16

[2] W. Narkiewicz, “Elementary and Analytic Theory of

Algebraic Integers,” PWN, Warszawa, 1990.

[3] H. Hasse, “Uber Mehrklassige Uaber Eigen Schlechtige

Reel-Quadratische Zahlkorper,” Elementary Mathematics,

Vol. 20, 1965, pp. 49-59.

[4] K. Szymiczek, “Knebush-Milnor Exact Sequence and

Parity of Class Numbers,” Ostraviensis Acta Mathe-

matica et Informatica Universitatis Ostraviensis, Vol. 4,

1996, pp. 83-95.

[5] A. Grytczuk and J. Grytczuk, “Some Results Connected

with Class Number in Real Quadratic Fields,” Acta

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HUdoi:10.1007/s10114-005-0544-2U

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doi:10.4064/aa107-2-5

[7] Z. Cao and X. Dong, “Diophantine Equations and Class

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[10] M. H. Le, “Divisibility of the Class Number of the Real

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14 ,







” Acta Mathematica Sinica,

Vol. 33, 1990, pp. 565-574.



[11] H. W. Lu, “The Divisibility of the Class Number of Some

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doi:10.3792/pjaa.76.91

[16] A. Grytczuk, “Some Remarks on Fermat Numbers,” Dis-

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