On Arithmetic Functions

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1 Globl ournl of Mthemticl Sciences: Theory nd Prcticl ISSN Volume 5, Number (0, 7- Interntionl Reserch Publiction House htt://wwwirhousecom On Arithmetic Functions Bhbesh Ds Dertment of Mthemtics, Guhti University, Guwhti-7804, Assm, Indi E-mil: Abstrct In this er, we intend to estblish some inequlities on ordn totient function Using rithmetic functions lie the Dedeind s function, the divisors function, Euler s totient function nd their comositions we gve some results on erfect numbers nd generlized erfect numbers Mthemtics Subject Clssifiction: A5, A4, Y70 Keywords: erfect number, -hyererfect number, suer-erfect number, suer-hyererfect number Introduction Let ( n denote the sum of divisors of e ( n d, where ( It is wellnown tht s sid to be erfect number if ( n n Euclid nd Euler showed tht even erfect numbers re of the form n (, where nd re rimes The rime of the form is clled Mersenne rime nd t this moment there re exctly 48 such rimes, which men tht there re 48 even erfect numbers [] On the other hnd, no odd erfect number is nown, but no one hs yet been ble to rove conclusively tht none exist In [5] the uthor rovided detiled informtion on erfect numbers Positive integer n with the roerty ( n n is clled lmost erfect number, while tht of ( n n, qusi-erfect number For mny result nd conjectures on this toic, see [4] DSurynryn [0] introduced the nottion of suer-erfect numbers A ositive integer s sid to be suer- erfect number if ( ( n n nd even suer-erfect numbers re of the form such tht re Mersenne rimes It is sed in this er nd still unsolved whether there re odd suer-erfect numbers Miloni nd Ber [8] introduced the concet of

2 8 Bhbesh Ds hyererfect number nd they conjecture tht there re hyererfect numbers for every A Beg nd K Fogrsi gve tble of hyererfect numbers in [] nd defined ositive integer n s hyererfect number if ( n n + They roved tht if n (, where is rime, then s hyererfect number Perfect numbers re hyer- erfect numbers If ( ( n n +, then s clled suer-hyererfect number In [] uthors conjectured tht ll suer-hyererfect numbers re of the form n, where nd re rimes The ordn s totient function nd Dedeind s function re resectively defined s ( n n ( nd ( n n (, where runs through the distinct rime divisors of n Let by convention (, ( Euler s totient functios defined s ( n n( ie ( n ( n All these functions re multilictive, ie stisfy the functionl eqution f ( mn f ( m f ( n for gcd( m, n In [9] the uthor defines -erfect, -erfect, -erfect numbers, where " " denotes comosition Bsic symbols nd nottions: ( n Sum of divisors of n, ( n Dedeind s rithmetic function, ( n ordn totient function, ( n Euler s totient function, b divides b, Œbb does not divide b, r{ n} set of distinct rime divisors of n Min Results Proosition ( b ( b, for ny, b,, with equlity only if, r{ } r{ b}, where r{ } denotes the set of distinct rime fctors of Proof: Let b q r, then ( b b ( ( ( q r, Œb q, q b rœ, r b Since (, so ( ( ( ( b b q r b ( b ( b, for ny, b If r{ } r{ b}, then b q hence ( b ( b Thus,, b qœ, q b nd

3 On Arithmetic Functions 9 Proosition If r{ } r{ b}, then ( b ( ( b for ny, b, Proof: Let q nd b r, where re the common rime fctors, nd q r{ } re such tht q r{ b}, so nd,, 0 nd r{ } r{ b} ( b b ( ( ( ( ( b q r q But, ( so, ( q q q q ( b ( ( b ( ( b ( ( b This imlies, q ( b ( ( b, for ny, b Proosition Let n, nd suose tht s -deficient number, then ( ( ( n n Proof: If s -deficient number, then ( n n Agin for ny n, ( n is n even number Using Proosition, we get ( ( ( ( Let u, n even number, then u { ( n} ( n ( u ( Therefore, ( ( ( n ( ( n Proosition 4 For ny n, ( ( n n Proof: First remr tht for ny n, ( n is even number nd ( n n Let ( n u, n eventeger Proceeding s Proosition, we obtin ( u u (, ie ( ( ( ( n { ( n} n n Proosition5 If s n even erfect number, then ( ( n ( n Proof: If s n even erfect number, then n (, where Mersenne rime nd ( n n, so ( ( n j ( n n j ( ( n is Proosition6 If n be the roduct of distinct even erfect numbers n, n, n r, then r ( ( n ( ( n ( n ( n r

4 0 Bhbesh Ds b c ( n ( n r n r ( ( n ( ( n ( ( n ( ( n r Proof: Let i i n nn nr, where ( re distinct erfect numbers with Mersenne rimes i ( i rimes, i,,, r ( i ( ( ( for ech i, so r ( n r ( ( n ( n ( n r i i b ( ( for ech i, nd clerly ( n r ( ( i n Therefore ( n ( n ( n ( n r n r ( n ( n ( n ( n r ( ( ( (, for echi, so( ( n ( r ( ( ( r = r = ( ( n ( ( n ( ( n r Proosition 7 If n be the roduct of distinct -hyererfect numbers n, n, n r, then r ( n ( ( n ( n ( n r r b ( n ( ( n ( n ( n r 4 Proof: Let i i n nn nr, where ( distinct -hyererfect numbers with rimes re i ( i,,, r i i ( (, for echi, so ( ( ( ( ( ( ( r r Then n r ( ( n ( n ( n r 4 (b i i ( ( ( ( ( r r ( n ( ( (, for ech i r ( ( n ( n ( n r 4 ( ( n Proosition8 If s suer-hyererfect number, then( ( n Proof: From definition of suer-hyererfect number we now tht n nd re rimes ( 4 n 4n, so ( ( n 4 nd 9, where ( n, so

5 On Arithmetic Functions 4n ( ( n ( ( n 4 Therefore ( ( n Proosition9 If n be the roduct of distinct suer-hyererfect numbers n, n, n r, then r ( ( n ( n ( n ( nr r r b ( ( n ( n ( n ( nr r Proof: Let n n n nr, where i re distinct suer-hyererfect numbers with i nd i re rimes ( i,,, r Agin ( i nd ( 4 i ( ( ( ( n ( r ( ( ( r ( ( ( r, so, r ( ( ( ( ( n r ( n ( n ( n r r 9 ( ( ( b ( n ( r 4 ( ( ( r ( ( ( r 4, so r 8 ( ( ( ( ( n r ( n ( n ( n r r Proosition0 ( There re infinitely mny n such tht ( ( n ( ( n (b ( ( n There re infinitely mny n such tht ( ( n (c There re infinitely mny n such tht ( ( n ( ( n n (d There re infinitely mny n such tht ( ( n n ( ( n Proof: ( is vlid for n for ny, since (, n ( ( n nd (, ( ( This imlies ( ( n ( ( n For (b ut n b, (, b Then ( b b, b b b 4 b b ( ( ( n Agin (, b b 8 ( ( n ( ( n, so ( ( n To rove (c ut n 7 b, b b b 576 b b (, b ( 7 7, ( ( 7 n, nd ( 7 7, 4

6 Bhbesh Ds b ( ( 7 n, So ( ( n ( ( n n To rove (d, ut n 5,( 576 (5 65, ( (5 n, 65 So, ( ( n n ( ( n 88 (5 45, ( (5 n, 5 Proosition ( There re infinitely mny n such tht ( ( n ( ( n ( ( n (b There re infinitely mny n such tht ( ( n Proof: ( is vlid for n for ny (b is vlid for n b for ny, b Then n esy comuttion shows tht ( ( n ( ( n n nd ( ( n n, so ( ( n 7 9 References: [] A Bege, K Fogrsi, Generlized erfect numbers, Act Univ Sientie, Mthemtic,,, ( [] LEDicson, History of the Theory of Numbers, Vol, Stechert New Yor 94 [] RGur, N Bircn, On Perfect Numbers nd their Reltions, ijcm, vol5, 00no7, 7-46 [4] RK Guy: Unsolved roblems in number theory, Third ed, Sringer Verlg, 004 [5] O Knill, The oldest oen roblem in mthemtics, NEU Mth Circle, December, 007 [6] Luquette, Perfect numbers, mthbuedu/eole/ost/teching/m4/oedf [7] S McCrnie, A study of hyererfect numbers, ournl of Integer Sequences, Vol(000 Article 00 [8] D Minoli, R Ber, Hyererfect numbers, Pi, Mu Esilon, 6 (975, 5-57 [9] Sndor: On the comosition of some rithmetic functions, Studi Bbes Bolyi University, Mth, 4 (989, 7-4 [0] DSurynryn, Suer erfect numbers, ElemMtg4 (969, 6-7 [] Gret Internet Mersenne Prime Serch (GIMPS

Quadratic Residues. Chapter Quadratic residues

Quadratic Residues. Chapter Quadratic residues Chter 8 Qudrtic Residues 8. Qudrtic residues Let n>be given ositive integer, nd gcd, n. We sy tht Z n is qudrtic residue mod n if the congruence x mod n is solvble. Otherwise, is clled qudrtic nonresidue