Some congruences related to harmonic numbers and the terms of the second order sequences

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1 Mathematca Moravca Vol. 0: 06, 3 37 Some congruences related to harmonc numbers the terms of the second order sequences Neşe Ömür Sbel Koaral Abstract. In ths aer, wth hels of some combnatoral denttes, we nvestgate varous basc congruences nvolvng harmonc numbers terms of the second order sequences {U n } {V n }.. Introducton The second order sequence {W n c, d; r, s}, or brefly {W n }, s defned for n > 0 by W n rw n sw n whch W 0 c, W d, where c, d, r, s are arbtrary ntegers. As some secal cases of {W n }, denote W n 0, ; r,, W n, r; r, by U n V n, resectvely. When r, U n F n the nth Fbonacc number V n L n the nth Lucas number. If α β are the roots of the equaton x rx 0, the Bnet formulas of the sequences {U n } {V n } have the forms U n αn β n α β V n α n β n, resectvely. From [, 3], E. Kılıç P. Stanca derved the followng recurrence relatons for the sequences {U n } {V n } for 0, n > 0. It s clearly that U n V U n U, V n V V n V, where the ntal condtons of the sequences {U n } {V n } are 0, U,, V, resectvely. Bnet formulas of the sequences {U n } {V n } are 000 Mathematcs Subject Classfcaton. Prmary: B39; Secondary: B50, 05A0, 05A9. Key words hrases. Congruences, Harmonc numbers, Second order sequences. 3 c 06 Mathematca Moravca

2 4 Some congruences gven by U n αn β n α β V n α n β n, resectvely. From the Bnet formulas, one can see that U n n U n U n U n V n. Harmonc numbers are those ratonal numbers gven by H 0 0, H n n, n N {,,... }. The frst few harmonc numbers are, 3, 6, 5,.... For m Z, harmonc numbers of order m are those ratonal H n,m n m, n N. For a rme an nteger a wth a, we wrte the Fermat quotent q a a /. Let Z be the set of ntegers. Z denote the set of those ratonal numbers whose denomnator s not dvsble by s called as the set of -adc nteger numbers. For an nteger D, D Z f D D / Z f D n [6]. It s clearly that x x has two smle roots n Z f only f ± mod. In [7], Z.W. Sun L.L. Zhao establshed arthmetc roertes of harmonc numbers. For examle, for any rme > 3, H 7 4 B 3 mod, where B 0, B, B,... Bernoull numbers. In [], A. Granvlle showed the congruence q x G x mod, > 3, where q x x x G x H, 3 8 B 3 mod, In [4], H. Pan Z. W. Sun showed the followng lemma rooston: Lemma.. Let > 3 be a rme. Then x. x x x x x mod.

3 Neşe Ömür Sbel Koaral 5 Prooston.. Let r s be nonzero ntegers. For an odd rme such that rs, y r r γ r γ 3 δ s mod, y r rγ γ r r γ 4 δ s mod, where y n W n, r; r, s γ, δ are the two roots of the equaton x rx s 0. In ths aer, we nvestgate the congruences nvolvng harmonc numbers terms of second order sequences {U n } {V n }. For examle, for, V U H V V H, V V / U 4 q V mod, V V mod, where V 4, a rme number > 3, an nteger wth V.. Some Lemmas In ths secton, we need the followng lemmas for further use. Lemma.. For n N x R, we have the followng sums: 5 6 x j H j j x j H j, j x x xn x H, Proof. For the roof of 5, from the sum x j H j j x j j j x x xn x H,. 0 j x y n xn y n x y, we have x j

4 6 Some congruences x j j0 j0 x j x n x x x x x n x x x xn x H, as clamed. Smlarly, the other result s roven. Thus, ths ends the roof. Lemma.. For n N x R, we have the followng sums: 7 8 jx j H j nxn x x x n x x H j xn x x x x x, jx j H j, nxn x x x n x x H, j Proof. For the frst clam, from the sums we wrte 0 0 x x x x x. x y n xn y n x y x y n nxn x y x x n y n x y, jx j H j j jx j j j j0 jx j j jx j jx j j0 nx n x x x n x

5 Neşe Ömür Sbel Koaral 7 x x x x x nxn x x x n x x H xn x x x x x, as clamed. The other clam s smlarly obtaned. Thus, the roof s comleted. Lemma.3. For n N x R, we have the followng sums: x nx n x n x x H x 3 H 9 x H, Proof. Consderng the sums 0 nxn x 3x n x x x x x x 3 x 3, x nx n x n x x 3 H, xn x x x x x x x x 3 x y n nxn x y x x n y n x y, x y n 0 0 x. x y n xn y n x y x nx n ny x xy xy n x y x y 3, the roof s clearly gven. Lemma.4. Let be an odd rme. For, 0 U 4 V 4 4 V U V V V U mod, mod,

6 8 Some congruences where V 4 Legendre symbol ṗ. Proof. For the roof of, usng the Bnet formula of the sequence {V n } tang α, β x nstead of x n β α x x mod [5], where any -adc nteger x. We get V 4 α 4 β β 4 α β 4 V β α V 4 α V 4 V 4 α V β α β α β V V U mod. Smlarly, usng Bnet formula of the sequence {U n }, the roof of the congruence n 0 s gven. Lemma.5. Let > 3 be a rme., V V Proof. Consder For V V V For an nteger wth V V V V α V V β α V α, tang α V, β V V V V V V β β mod. V lace of x n, resectvely, we wrte.

7 V α V α V β V Neşe Ömür Sbel Koaral 9 β V β V α V α β α β α β V V V V V V as clamed. mod, Lemma.6. Let > 3 be a rme., For an nteger wth V Proof. Consder that V V V V V V V mod. α V α V V β V β V. For, by tang V, nstead of r, s n 4, resectvely, we have V V V V β V α V β V mod, from Fermat s lttle theorem, the congruence α β, we get mod for 3 β β V β V α β β V V V α α β β β V mod. V β By 3, we obtan the desred result.

8 30 Some congruences 3. The Results nvolvng the terms of the sequences {U n } {V n } In ths secton, we gve congruences for the terms of the sequences {U n } {V n }. Now we start wth our frst result. Theorem 3.. Let be an odd rme. For, 4 5 V 4 V U H V V / U q V mod, V H V U V q U mod, where the Fermat quotent q /. Proof. For the roof of 4, by the Bnet formula of the sequence {U n }, we have V U H V α H V β H. Wrtng / lace of n α 4, β 4 lace of x n 5, resectvely, we wrte α β 4 4 α 4 H β 4 H α 4 β 4 Snce α β α 4 β α β 4, we can rewrte α H, β H. 6 7 V V α 4 H β 4 H α 4 β 4 α H, β H.

9 Neşe Ömür Sbel Koaral 3 By 6 7, we get V α 4 V 4 U H α H V H, β 4 β H whch, by the congruence H q mod, equvalents V V U 4 q V mod. Smlarly, usng the Bnet formula of the sequence {V n }, 6, 7, 0 the congruence H q mod, the other clam s obtaned. For examle, by tang n Teorem 3., for r 4, r r 4 U 4 H r V r 4 / U 4 q V mod, r V 4 H r U r 4 V q U Theorem 3.. Let be an odd rme. For, mod. 8 V V U 4 H U V V q U V U V mod, V 4 H V V U q V

10 3 Some congruences 9 4 V V / U mod, where q as before. Proof. For the roof of 9, usng the Bnet formula of the sequence {V n }, we have V V 4 H V α 4 H V β 4 H. Puttng / nstead of n α 4, β 4 nstead of x n 7, resectvely, we wrte α 4 α 4 H α α 4 α α β 4 β 4 H α 4 β β 4 β β β 4. H H From the equaltes α β α 4 β α β 4, we have 0 V V α 4 H β 4 H V α α H α α 4 V β β H β β 4,.

11 Neşe Ömür Sbel Koaral 33 Usng the Bnet formulas of the sequences {U n } {V n }, by 0, we rewrte V V 4 H V α α H α 4 α V β β H β β 4 V U V H V From the congruence H q mod, we have V V 4. V 4 H V V U q V 4 V V U mod, as clamed. Smlarly, the other congruence s gven. Thus, the roof s comleted. For examle, when r n Teorem 3., we have the congruences as follows: For 5, 5 5 F 4 H F L q F F 5 L mod, L 4 H L 5F q L L 5 / F 4 mod.

12 34 Some congruences Theorem 3.3. Let be an odd rme. For, V 3 U 4 H V 3 4 V V U V4 V 3 q 3 U U mod, V 3 U 3 q V 4 H V V U V4 3 U V V mod, Proof. Usng the Bnet formulas of the sequences {U n }, {V n }, by 9, 0 the congruence H q mod, we obtaned the desred result. Now, we wll gve the congruences wth harmonc numbers of order, H n,. Theorem 3.4. Let > 3 be a rme. For, V V H, V V mod. Proof. From Bnet formula of the sequence {V n }, we consder α V β V V V H, α V α V α V H, H, β V H, α V β V H, β V β V H, H,.

13 Neşe Ömür Sbel Koaral 35 By tang nstead of n α V, β V V α V V β 3 V α H, V β H, V nstead of x n 6, resectvely, we have α V α β V V H,, β H,. From, 3 the congruence H, 0 mod, we get V α V H, V α V V mod. Usng Bnet formula of the sequence {V n } Lemma.6, we have whch settles the roof. V V H, V V V V mod. Theorem 3.5. Let > 3 be a rme. For an nteger wth V, V 3 V H, V V V V V V V mod. Proof. From Bnet formula of the sequence {V n } α β, we have V 3 V H, α 3 V α V β V H, β 3 V H, α V H, β V α V H, α V β V H, β V H,.

14 36 Some congruences If we tae nstead of n α V, β V 4 5 β V α V V α V β α α V α V H, nstead of x n 8, resectvely, we get α α α V β V V β V α β β V H, α V, V β β β V β V. V H, H, From 4, 5 the congruence H, 0 mod, we have V 3 V H, β α α V By α β, we rewrte V 3 V H, α β β V α V α V β V mod. α V β β V V V whch, by Lemma.5 Lemma.6, equvalents V V V V V V V V V V mod, mod.

15 Neşe Ömür Sbel Koaral 37 As a result of Teorem 3.5, by tang nstead of, we have the followng corollary: Corollary 3.. Let > 3 be a rme. For r,, V 3 r H, r V V r r V r mod. For examle, when r n Corollary 3., we have the congruence as follows:for, L 3 H, L L 3 References L mod. [] A. Granvlle, The square of the Fermat quotent, Integers:Electron. J. Combn. Number Theory, 4 004, A. [] E. Kılıc, P.Stanca, Factorzatons reresentatons of second lnear recurrences wth ndces n arthmetc rogressons, Bulletn of the Mexcan Mathematcal Socety, Vol. 5, No. 009, [3] E. Kılıc, P.Stanca, Factorzatons of bnary olynomal recurrences by matrx methods, Rocy Mountan Journal of Mathematcs, Vol. 4, No. 4 0, [4] H. Pan, Z. W. Sun, Proof of three conjectures on congruences, Sc. Chna Math., Vol. 57, No. 0 04, [5] S. Koaral N. Ömür,On congruences related to Central bnomal coeffcents, Harmonc Lucas Numbers, Tursh Journal of Mathematcs, acceted. [6] P.T. Young,-adc congruences for the generalzed Fbonacc sequences, The Fbonacc Quart., Vol. 3, No. 994,. -0. [7] Z. W. Sun, L. L. Zhao, Arthmetc theory of harmonc numbers II, Colloq. Math., Vol. 30, No. 03, Neşe Ömür Kocael Unversty Mathematcs Deartment 4380 İzmt Kocael Turey E-mal address: neseomur@ocael.edu.tr Sbel Koaral Kocael Unversty Mathematcs Deartment 4380 İzmt Kocael Turey E-mal address: sbel.oaral@ocael.edu.tr