Fermat Numbers in Multinomial Coefficients

Transcription

1 Joural of Ieger Sequeces, Vol. 17 (014, Aricle Ferma Numbers i Muliomial Coefficies Shae Cher Deparme of Mahemaics Zhejiag Uiversiy Hagzhou, Chia chexiaohag9@gmail.com Absrac I 001 Luca proved ha o Ferma umber ca be a orivial biomial coefficie. We exed his resul o muliomial coefficies. 1 Iroducio Le F m = m +1 be he m h Ferma umber for ay oegaive ieger m. Several auhors sudied he Diophaie equaio = m +1 = F m, (1 k where k, ad m 0. We refer o he aricles [, 3, 5, 6, 8] for furher deails. I 001, Luca [6] compleely solved Eq. (1 ad proved ha i has oly he rivial soluios k = 1, 1 ad = F m. The proof is maily based o a cogruece give by Lucas [7]. For more abou Ferma umbers, see [4]. For a posiive ieger, le, be oegaive iegers, ad defie he -order muliomial coefficie as follows: = ( 1 ( k 1 k +1, k 1! k! wih i=1 k i < +1. I paricular, ( 0,...,0 = 1. Noe ha for, if i=1 k i =, he he -order muliomial coefficie equals a ( 1-order muliomial coefficie =. 1 1

2 There are may papers cocerig he Diophaie equaios relaed o muliomial coefficies. For example, Yag ad Cai [9] proved ha he Diophaie equaio = x l has o posiive ieger soluios for, 3, l, ad i=1 k i =. I his paper, we cosider he Diophaie equaio ad prove he followig heorem. = m +1 = F m, for, ad k i <, ( Theorem 1. The Diophaie equaio ( has o ieger soluios (m,, for oegaive m ad posiive,. i=1 Two Lemmas To prove Theorem 1, we eed he followig wo lemmas. Lemma (Euler [1]. Ay prime facor p of he Ferma umber F m saisfies p 1 (mod m+1. Lemma 3 (Luca [6]. If F m = s k, wih m 5, s 1, ad 1 k, we have he followig wo properies. (i Le = d, where A = { p : prime p, ad p 1 (mod m+1 }, ad = p Ap αp. The k = d < m. (ii k i i for ay i = 0,...,k 1. Remark 4. Lemma 3 is summarized from Luca s proof [6] of Diophaie equaio (1. Alhough Luca oly proved he case s = 1, he idicaed ha he resul also holds for all posiive iegers s.

3 3 Proof of Theorem 1 The firs five Ferma umbers are primes, which cao be a muliomial coefficie i Eq. (. Therefore, we oly eed o cosider m 5. Moreover, for ay muliomial coefficie wih > 0, k1,...,k 1, ad i=1 k i <, here exiss a muliomial coefficie such ha 1 k 1,...,k. Hece, Eq. ( becomes Le = d, where =, k 1,...,k F m =, for m 5, 1 k i, ad k i <. (3 i=1 A = { p : prime p, ad p 1 (mod m+1 }, ad = p Ap αp. For ay i = 1,...,, we have k i F m = =, (4 k i k 1,...,k i 1,k i+1,...,k where ( k i k 1,...,k i 1,k i+1,...,k is a posiive ieger. By Lemma 3 (i, we have ki = d < m for i = 1,...,. The Eq. (3 becomes F m =, > d, ad (5 d,...,d d d =. (6 d d d,...,d Noe ha d 1. We sudy Eq. (6 i he followig hree cases. Case 1: d >. Sice > d ad d, we have 3d. The, d d <. 3

4 I Eq. (6, applyig Lemma 3 (ii o ( d ad d d respecively, ad seig i = 1, we have d 1 1 ad Thus, d 1 d, which is impossible. d 1 d 1. Case : d =. Le =. The Eq. (5 becomes F m = = ( 1( 1( 4 3.,...,,..., The ad 1 are boh F m s facors. Accordig o Lemma, we obai 1 1 (mod m+1, which is impossible. Case 3: d = 1. Eq. (5 becomes F m = 1,...,1 = ( 1. 1,...,1 The ad 1 are boh F m s facors. Accordig o Lemma, we obai 1 1 (mod m+1, which is also impossible. This complees he proof of Theorem 1. Remark 5. Oe ca eve fid ha he muliomial coefficie i Eq. ( could o divide a Ferma umber. Oherwise, assume ha here exiss a posiive ieger s such ha F m = s. Noe ha i Eq. (4 we sill have F m, k i ad i Eqs. (5 ad (6 similar resuls hold. Hece, we ca ge he proof i he same way. 4 Ackowledgmes I am idebed o Mr. Yog Zhag for providig releva refereces ad examiig he whole proof, ad o Mr. Jiaxig Cui for givig deailed commes. I am also idebed o he referee for his careful readig ad helpful suggesios. 4

5 Refereces [1] L. Euler, Observaioes de heoremae quodam Fermaiao aliisque ad umeros primos specaibus, Acad. Sci. Peropol. 6 (1738, Available a hp://eulerarchive.maa.org/pages/e06.hml. [] D. Hewgill, A relaioship bewee Pascal s riagle ad Ferma s umbers, Fiboacci Quar. 15 (1977, [3] H. V. Krisha, O Mersee ad Ferma umbers, Mah. Sude 39 (1971, [4] M. Křížek, F. Luca, ad L. Somer, 17 Lecures o Ferma Numbers: From Number Theory o Geomery, Spriger, 001. [5] F. Luca, Pascal s riagle ad cosrucible polygos, Uil. Mah. 58 (000, [6] F. Luca, Ferma umbers i he Pascal riagle, Divulg. Ma. 9 (001, [7] É. Lucas, Théorie des focios umériques simpleme périodiques, Amer. J. Mah. 1 (1878, , , [8] P. Radovici-Mărculescu, Diophaie equaios wihou soluios(romaia, Gaz. Ma. Ma. Iform. 1 (1980, [9] P. Yag ad T. Cai, O he Diophaie equaio ( k 1,...,k s = x l, Aca Arih. 151 (01, Mahemaics Subjec Classificaio: Primary 11D61; Secodary 11D7, 05A10. Keywords: Ferma umber, muliomial coefficie. (Cocered wih sequece A Received Jauary ; revised versio received February Published i Joural of Ieger Sequeces, February Reur o Joural of Ieger Sequeces home page. 5