MATHEMATICA MONTISNIGRI Vol XXXVIII (017) MATHEMATICS CERTAIN CONGRUENCES FOR HARMONIC NUMBERS ROMEO METROVIĆ 1 AND MIOMIR ANDJIĆ 1 Maritie Faculty Kotor, Uiversity of Moteegro 85330 Kotor, Moteegro e-ail: roeo@ac.e Faculty for Iforatio Techology, Uiversity Mediterraea 81000 Podgorica, Moteegro e-ail: ioir.adjic@uieditera.et Suary. For give positive itegers ad, the haroic ubers of order are those ratioal ubers H, defied as H, 1 =. = 1 If =1, the H : = H,1 = 1 is the th haroic uber. I [1] Z.W. Su obtaied = 1 basic cogrueces odulo a prie p > 3 for several sus ivolvig haroic ubers. Further geeralizatios ad extesios of these cogrueces have bee obtaied by R. Tauraso i [16], by Z.W. Su ad L.L. Zhao i [14] ad by R. Me_strovi_c i [6] ad [7]. I this paper we prove that for each prie p > 3 ad all itegers = 0, 1,,p there holds = ( 1) p H 1 ph + 1 + H + 1 H + 1, od p + 1 3 As a applicatio, we deterie the od p 3 cogrueces for the sus r = 0, 1,, 3 ad a prie p > 3. r H = 1 with 1 INTRODUCTION Give positive itegers ad, the haroic ubers of order are those ratioal ubers H, defied as For siplicity, we will deote by H H, 1 =. = 1 : = H,1 = = 1 1 010 Matheatics Subject Classificatio: Priary 11A07; Secodary 05A10, 05A19, 11B65, 11B68. Key words ad Phrases: haroic uber, haroic ubers of order, cogruece odulo a prie (prie power), Wolstehole's theore. 5
ROMEO MEŠTROVIĆ AND MIOMIR ANDJIĆ the th haroic uber (i additio, we defie H 0 = 0). Haroic ubers play iportat roles i atheatics. Throughout this paper, for a prie p ad two reduced ratioal ubers a/b ad c/d such that b ad d are ot divisible by p, we write a/b c/d(od p s ) (with s N) to ea that ad bc is divisible by p s. I 01 Z.W. Su [1] ivestigated their arithetic properties ad obtaied various basic cogrueces odulo a prie p > 3 for several sus ivolvig haroic ubers. I particular, Su established the cogrueces =1 (H ) r (od p 4 r ) for r = 1,, 3. Further geeralizatios ad extesios of these cogrueces have bee obtaied by R. Tauraso i [16], by Z.W. Su ad L.L. Zhao [14] ad by R. Meštrović i [6] ad [7]. Furtherore, Z.W. Su [13] iitiated ad studied cogrueces ivolvig both haroic ad Lucas sequeces (especially, icludig Fiboacci ubers or Lucas ubers). Moreover, soe cogrueces ivolvig ultiple haroic sus were established i [9], [18] ad [19]. Recall that Beroulli ubers B 0, B 1, B,... are recursively give by + 1 B 0 = 1 ad B = 0 ( = 1,, 3,...). =0 It is easy to fid the values B 0 = 1, B 1 = 1, B = 1 6, B 4 = 1 30, ad B = 0 for odd 3. Furtherore, ( 1) 1 B > 0 for all 1. These ad ay other properties ca be foud, for istace, i [3]. Recetly, the first author of this paper i [6, Theore 1.1] established the followig six cogrueces ivolvig haroic ubers cotaied i the followig result. Theore 1.1 ([6, Theore 1.1]). Let p > 5 be a prie, ad let q p () = ( 1)/p be the Ferat quotiet of p to base. The ad =1 H =1 =1 =1 q p () + 3 pq p() 3 + p 1 B p 3 (od p ), (1) H 1 3 q p() 3 + 3 4 B p 3 (od p), () =1 H =1 H, I this paper we prove the followig result. H 5 8 B p 3 (od p), (3) 1 3 q p() 3 + 11 4 B p 3 (od p), (4) H 7 8 B p 3 (od p) (5) 1 3 q p() 3 5 4 B p 3 (od p). (6) 6
ROMEO MEŠTROVIĆ AND MIOMIR ANDJIĆ 3 Theore 1.. Let p > 3 be a prie. The for each = 0, 1,..., p there holds )H (1 ( 1) ph +1 + p + 1 (H +1 H +1, ) (od p 3 ). (7) = The particular cases of Theore 1. yield the followig result. Corollary 1.3. Let p > 3 be a prie. The H 1 p (od p 3 ), (8) =1 ad H p 3p + 4 =1 H 15p 17p + 6 36 =1 3 H 1p 10p 48 =1 (od p 3 ), (9) (od p 3 ), (10) (od p 3 ), (11) Reducig the odulus i cogrueces (8), (9), (10) ad (11) of Corollary 1.3, iediately gives the followig two corollaries. Corollary 1.4. Let p > 3 be a prie. The H 1 p (od p ), (1) =1 ad H 3p, (od p ), (13) 4 =1 H 17p 6 36 =1 3 H 5p 4 =1 (od p ), (14) (od p ). (15) Corollary 1.5. Let p > 3 be a prie. The H 1 (od p), (16) =1 7
4 ROMEO MEŠTROVIĆ AND MIOMIR ANDJIĆ H 1 =1 (od p), (17) H 1 6 =1 (od p), (18) ad 3 H 0 (od p). (19) =1 Rear 1.5. Notice that the cogrueces (8) ad (9) are proved by Z.W. Su i [1, p. 419 ad p. 417]. PROOF OF THEOREM 1. AND COROLLARY 1.3 For the proof of Theore 1. we will eed the followig three auxiliary results. Lea.1 (see the idetity (6.70) i [1]; also [11, p. ]). If ad are oegative itegers such that, the = H = ( + 1 H +1 1 ). (0) + 1 + 1 The followig result is well ow as Wolstehole s theore established i 186 by J. Wolstehole [17]. Lea. (see [17]; also [], [5] ad [10]). If p > 3 is a prie, the H 0 (od p ). (1) The followig result is well ow ad eleetary. Lea.3 (see, e.g., [1, Lea.1 (.)]). If p 3 is a prie, the ( p 1 ) ) ( 1) (1 ph + p (H H, ) (od p 3 ), () for each = 0, 1,..., p 1. 8
ROMEO MEŠTROVIĆ AND MIOMIR ANDJIĆ 5 Proof of Theore 1.. Taig = p 1 ito the idetity (0) of Lea.1 ad usig the idetities ( p +1 = p ) ( +1 ad p ) ( +1 ) ( = +1), we fid that ( p H = H p 1 ) + 1 + 1 = ( p = H + 1 + 1 p 1 ) + 1 p = H + 1 p 1 p (3) + 1 p + 1 + 1 + 1 p = H + 1 p 1 1 p + 1 + 1 + 1 + 1 p = H 1 ( ) p p 1 + 1 + 1 + 1 p p 1 = H 1 p 1. + 1 + 1 + 1 Usig the cogruece (1) of Lea. ad the assuptio 0 p, we obtai p p 1 H 0 (od p 3 ). (4) + 1 Furtherore, by the cogruece () of Lea.3, we have ) p 1 ( 1) (1 ph +1 + p + 1 (H +1 H +1, ) (od p 3 ). (5) Applyig the cogrueces (4) ad (5) to the right had side of the idetity (3), we iediately get )H (1 ( 1) ph +1 + p + 1 (H +1 H +1, ) (od p 3 ). (6) = The cogruece (6) is actually the cogruece (7) of Theore 1.. This copletes the proof. Proof of Corollary 1.3. Taig = 0 ad = 1 ito the cogruece (7) of Theore 1., we iediately give the cogrueces (8) ad (9), respectively. Taig = ito the cogruece (7), we fid that H 1 3 = which ca be writte as = H = H ( 1 11p 6 + p 1 3 ) ( 1 11p ) 6 + p (od p 3 ), (od p 3 ). (7) 9
6 ROMEO MEŠTROVIĆ AND MIOMIR ANDJIĆ By usig the cogrueces (7) ad (9), we obtai H H + ( 1 11p ) 3 6 + p =1 =1 p 3p + 4 + 3 ( 1 11p 6 + p (od p 3 ) ) (od p 3 ) = 15p 17p + 6 (od p 3 ). 36 The above cogruece is i fact the cogruece (10) of Corollary 1.3. Fially, i order to prove the cogruece (11), we put = 3 ito the cogruece (7). This iediately yields H 1 ( 1 5p ) 3 4 1 + 35p (od p 3 ). 4 =3 By substitutig 3 = 3 3 + ito above cogruece, it ca be writte as 6 3 H H H + 1 ( 1 5p ) 6 3 4 1 + 35p (od p 3 ). (8) 4 =3 =3 =3 By usig the cogrueces (8), (9) ad (10), we have 3 H 3 H H 3 ( 1 5p ) 1 + 35p 4 =1 =1 15p 17p + 6 1 = 1p 10p =1 + p 3p + (od p 3 ). 35p 50p + 4 16 (od p 3 ) (od p 3 ) 48 The above cogruece coicides with the cogruece (11) of Corollary 1.3, ad thus, the proof is copleted. Rear.4. Of course, by applyig the recursive ethod used i proof of Corollary 1.3, it is possible to deterie the expressio for =1 H (od p 3 ) for each positive iteger, where p > 3 is a prie. Furtherore, it is obvious that each of these cogrueces ca be writte i the for H a p + b p + c (od p 3 ), =1 where a, b ad c are ratioal ubers depedig o whose deoiators are ot divisible by p. 10
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