Sandor J. - Geometric Theorems, Diophantine Equations and Arithmetic Functions.pdf - Number Theory - annam101

József Sándor

G EOMETRIC THEOREMS , DIOPHANTINE

EQUATIONS , AND ARITHMETIC FUNCTIONS

*************************************

AB/AC=(MB/MC)(sin u / sin v)

1/z + 1/y = 1/z

Z(n) is the smallest integer m

such that 1+2+…+m is divisible by n

*************************************

American Research Press

Rehoboth

2002

József Sándor

D EPARTMENT OF M ATHEMATICS

B ABE Ş -B OLYAI U NIVERSITY

3400 CLUJ - NAPOCA , ROMANIA

Geometric Theorems, Diophantine Equations, and

Arithmetic Functions

American Research Press

Rehoboth

2002

This book can be ordered in microfilm format from:

Books on Demand

ProQuestInformationandLearning

(UniversityofMicrofilmInternational)

300N.ZeebRoad

P.O. Box 1346, Ann Arbor

MI48106-1346,USA

Tel.:1-800-521-0600(CustomerService)

http://wwwlib.umi.com/bod/

Rehoboth, Box 141

NM 87322, USA

E-mail: M_L_Perez@yahoo.com

More books online can be downloaded from:

http://www.gallup.unm.edu/~smarandache/eBook-otherformats.htm

Referents: A. Bege , Babeş-Bolyai Univ., Cluj, Romania;

K. Atanassov , Bulg. Acad. of Sci., Sofia,

Bulgaria;

V.E.S. Szabó , Technical Univ. of Budapest,

Budapest,Hungary.

ISBN : 1-931233-51-9

Standard Address Number 297-5092

Printed in the United States of America

"...It is just this, which gives the higher arithmetic that magical charm which has made

it the favourite science of the greatest mathematicians, not to mention its inexhaustible

wealth, wherein it so greatly surpasses other parts of mathematics..."

(K.F. Gauss, Disquisitiones arithmeticae, Gottingen, 1801)

Preface

This book contains short notes or articles, as well as studies on several topics of

Geometry and Number theory. The material is divided into ve chapters: Geometric the-

orems; Diophantine equations; Arithmetic functions; Divisibility properties of numbers

and functions; and Some irrationality results. Chapter 1 deals essentially with geometric

inequalities for the remarkable elements of triangles or tetrahedrons. Other themes have

an arithmetic character (as 9-12) on number theoretic problems in Geometry. Chapter 2

includes various diophantine equations, some of which are treatable by elementary meth-

ods; others are partial solutions of certain unsolved problems. An important method is

based on the famous Euler-Bell-Kalmar lemma, with many applications. Article 20 may

be considered also as an introduction to Chapter 3 on Arithmetic functions. Here many

papers study the famous Smarandache function, the source of inspiration of so many

mathematicians or scientists working in other elds. The author has discovered various

generalizations, extensions, or analogues functions. Other topics are connected to the com-

position of arithmetic functions, arithmetic functions at factorials, Dedekind's or Pillai's

functions, as well as semigroup-valued multiplicative functions. Chapter 4 discusses cer-

tain divisibility problems or questions related especially to the sequence of prime numbers.

The author has solved various conjectures by Smarandache, Bencze, Russo etc.; see espe-

cially articles 4,5,7,8,9,10. Finally, Chapter 5 studies certain irrationality criteria; some of

them giving interesting results on series involving the Smarandache function. Article 3.13

(i.e. article 13 in Chapter 3) is concluded also with a theorem of irrationality on a dual

of the pseudo-Smarandache function.

A considerable proportion of the notes appearing here have been earlier published in

Sandor J. - Geometric Theorems, Diophantine Equations and Arithmetic Functions.pdf

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