On Sum of Powers of Numbers Having a Given Order Modulo a Power of a Prime

Transcription

1 On Sum of Powers of umbers Having a Given Order Modulo a Power of a Prime Yuguang Fang Department of Electrical and Computer Engineering niversity of Florida 45 Engineering uilding POox 6 Gainesville Florida 26-6 Tel: Fax: fang@eceufledu Proposed running title: On Sum of Powers of umbers Having a Given Order

2 Abstract Let denote the sum of th powers of numbers having given order or exponent modulo where is an odd prime and are positive integers and with indicating the Euler function n this paper we study the congruence property of this summation and obtain the following result! # $% & ' +*- */ eywords: Sum of umbers Residue Congruence Primitive roots AMS Subject Classification: A A25 ntroduction Let denote the sum of th powers of numbers having given order or exponent modulo where is an odd prime and are positive integers and with indicating the Euler function Gauss proved in his masterpiece [4] that :; where is the Möbius function n 8 Stern [6] generalized this result and obtained that < 2= %>89? for A ' n 88 Forsyth [] studied the congruence of for any positive integer however his results and proofs were very complicated n 952 Moller [2] investigated more general cases and obtained C 2 However Moller s proof was still complicated primitive root D>89E!< Gupta [5] gave a simpler proof using the concept of n this paper we generalize Moller s results to the case when the modulo is a power of a prime 2 Main Results The following are our main results Theorem Let and be positive integers let be a prime number is the number satisfying F $ and 2 then we have G+89H Let F let J L 8CMOP Q denote the highest power of in where 8CMRPTS' 2

3 A H A denotes the highest power of factor in S J For We have Theorem 2 define P 8CM! #$%'& 2 To prove our main results we need the following lemmas *+ Lemma There exists a primitive root modulo such that for any 8 and ' -/ 2 +89A 4265 Proof: Suppose is a primitive root modulo without loss of generality we assume J+89 5 where 9 t is well-known [5] that is also a primitive root modulo ; When from the choice of we know Lemma is true Suppose that Lemma is true for that is we have then * *4:; </ 5 -> 42 y induction we conclude that Lemma is true Lemma 2 [5] LetC TS' -/ 5 & 5 => H 265 denote an arithmetical function then 8DOTS' E#FGH C ML D 9L where S represents S and CQS' Lemma [5] O P 8CM?%? RQ J C C 8CM! ; 5 5 C TS' TS TS Q *- Lemma 4 [] Given integers and such that * and J elements in the set /V XW V< F! which are relatively prime to is QF then the number of

4 V ; V ; ow we are ready to prove our main results Proof of Theorem : Let be the primitive root as in Lemma setv thusv ; ; +89A D +89A! and have the same order Set t is observed that every element in J V W LPD! < J VE W LPD will reappear many times in which has the order then the number of elements in the set J V W V is equal to the number of elements in the set E +89H! L D! J W4 ++89! L D then we have modulo ; LetVE ; ; be any element in which is equal to JQF via Lemma 4 Thus every element in times in modulo will reappear exactly QF Let JMW L D then we have in what follows we will use to denote the congruence with respect to modulo for brevity C From Lemma 2 we have Let Then we Define! J 5 QTS' ; ; ; Q Q 2S 2S 4 ; *- $# # &% O ; ; ; ' ; ' 4 5

5 A For * if then we have +89/ ie ; ; is a primitive root modulo then we have? F ie : However since hence ; +89 Similarly we can obtain ; +89! L- L- Taking these two equations into Eq5 we obtain ; ; Q +89J L <L and ; we have Therefore for Applying Lemma we can obtain the following 8CM?%? n fact we only need to prove or 8CM?? 8CM which is obvious by noticing that where 8CM?? 8 8CM S- L S S 8CM 8CM ecause we then only need to prove 8CM S Thus we obtain the proof of Eq From Lemma there exists a *- with ; Thus we have ;! and Similarly we can obtain Taking Eq 8 and Eq 9 into Eq 6 we obtain Q ; ; ; Q such that :! :; = * ! ; +89J -> >89 +89? Since F we must have 6 % 8 *- 9 G+89<

6 Taking this into Eq we finally obtain C D D+89 J J G+89 This completes the proof of Theorem and C 2 F J J we have G+89 which is exactly the result obtained by Moller [2] Proof of Theorem 2: otice that is multiplicative and J is additive therefore is multiplicative Moreover it can be easily shown that Suppose that is a prime when we have ; Q Q f ; f ; Therefore if L L ; ; then ; ; O ' then ; This completes the proof of Theorem 2 ; J ' ; J 2 J 2 ; When is multiplicative in ' ' % J C? is the canonical prime factorization then we have ; ' ; R ' P ' otherwise 6

7 References [] T Apostal An ntroduction to Analytic umber Theory Springer-Verlag ew York 96 [2] R Moller Sums of powers of numbers having given exponent modulo a prime Amer Math Monthly [] AR Forsyth Primitive roots of primes and their residues Messager of Mathematics X [4] CF Gauss Disquisitiones Arithmeticae Springer-Verlag ew York-erlin 986 [5] H Gupta Selected Topics in umber Theory AACS Press 98 [6] MA Stern emerkungen über hohere arithmetik Journal für Mathematik V 8 4-5