Preface

Number theor y i s one of the few areas of mathematics fo r which most problem s

can b e understoo d b y just abou t anyone , o r a t leas t b y al l those wh o ar e familia r

with very basic notions of algebra, combinatorics and analysis. Ever y teacher know s

the importance of practicing problem solving : indee d i t turn s out t o be a great wa y

to learn how to reason, no matter th e area of mathematics the problems come from .

Number theor y i s quite appropriat e fo r thi s kin d o f exercise . Fo r thes e reasons , a

collection of problems in elementary or classical number theory seems in our opinio n

to b e a complementar y pedagogica l too l fo r an y learnin g proces s i n mathematics .

Moreover, a clever choice of problems can greatly help to raise the curiosity of those

who try t o solv e them .

Unfortunately, ver y fe w book s ar e entirel y dedicate d t o problem s i n numbe r

theory. Thes e includ e th e classica l wor k o f the grea t maste r W . Sierpinski entitle d

250 Problems in Elementary Number Theory an d publishe d i n Varsovi e i n 1970, a

book which is not well known and unfortunately ou t of print. Hence , our manuscrip t

does fill an importan t ga p in this are a an d moreove r i t ha s the advantag e o f havin g

been written to reach a large audience. On e can also see it as a practical complemen t

of a n earlie r boo k o f th e authors , tha t i s Introduction a la theorie des nombres

published b y MODUL O (2n d edition , 1997), o r t o an y othe r introductor y boo k i n

number theory .

Nevertheless, w e mus t admi t tha t ou r mai n motivatio n fo r writin g thi s boo k

has been ou r passio n fo r numbe r theory , namel y thi s branc h o f mathematics whic h

distinguishes itsel f b y it s beaut y an d it s numerou s mysteries , b y it s simplicit y an d

its complexity , tha t i s from th e proo f tha t ther e ar e infinitel y man y prime s t o th e

recently establishe d proo f o f Fermat's Las t Theorem .

This boo k obviousl y contain s man y problem s fro m elementar y numbe r theory .

Some of these are well known and can be found her e and there in introductory book s

in number theory, while others are not so common. Thi s is namely the case of several

problems which we picked from the lesser known manuscript o f Sierpinski mentione d

above. Ou r book also contains some problems submitted to the readers of three well

known journals: American Mathematical Monthly, Mathematics Magazine an d The

College Mathematics Journal. Finally , ou r boo k contain s som e 30 0 ne w problem s

never publishe d before .

The choic e of problems i s obviously subjective ; hence , it i s no coincidence tha t

the sectio n o n arithmetica l function s i s th e longest ! I n an y event , a n effor t ha s

been mad e t o cover , o r a t leas t brush , eac h o f th e classica l theme s o f elementar y

number theory . O n th e othe r hand , sinc e mor e an d mor e student s no w hav e t o

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