Note on the matrix Fermat s equation 1

Transcription

1 NNTDM 7 0,, 4- Note o the matrix Fermat s equatio Aleksader Grytczuk ad Izabela Kurzyd lo Faculty of Mathematics, Computer Sciece ad Ecoometrics, Uiversity of Zieloa Góra, Zieloa Góra, Polad s: {AGrytczuk, IKurzydlo}@wmieuzzgorapl Abstract We cosider the Fermat s equatio X Y = Z F i the set of ratioal matrices We give some ecessary coditio of solvabillity of this equatio Keywords The matrix equatio, the matrix Fermat s equatio, powers of matrices AMS classificatio 5A4,5A4 Itroductio The Fermat s equatio F i M Q was cosidered by Barett ad Weitkamp i 96 what was described by P Ribeboim i moograph [3] I 966 R Z Domiaty [4] discovered that the equatio F has ifiitely may solutios i M Z for = 4 The solvability of F i GL Z was first ivestigated by L N Vaserstei [4] A Khazaov i [9] gave ecessary ad sufficiet coditios for solvability F for X, Y, Z belogig to SL Z,- SL 3 Z, GL 3 Z A Gryczuk [7] proved some ecessary coditio to satisfy F i itegral matrices X, Y, Z, ad i [5] he gave a extesio of this result Studies coected with Khazaov s results effected too H Qi [] The equatio of Fermat was ivestigated by Z Patay ad A Szakas []}, Z Cao ad A Grytczuk [] I [3] Z Cao ad A Grytczuk gave a ecessary ad sufficiet coditio for solvability F for X, Y, Z SL Z For X = A x, Y = A y, Z = A z we obtai from F the followig equatio A x A y = A z The ecessary ad sufficiet coditios for solvabitity of the equatio i atural umber x, y, z ad >, where A M Z were give by Le ad Li i [0] Aother proof of this result was gave by A Grytczuk [6] I the paper [8] we cosidered the extesio of the equatio This paper is partly supported by EFS Europea Social Fuds 4

2 A mx A my A mz = A mw, where A M ZWe gave [8] the ecessary ad sufficiet coditios for solvabitity of the equatio i atural umber x, y, z, w ad > I this paper we give the followig ecessary coditio for solvabillity of the Fermat s equatio F i the set of ratioal matrices M Q : Theorem Let X, Y, Z M Q Let det X = det Y = k, det Z =,where k, k, 0, ad Z X = XZ, Z Y = Y Z, If the Fermat s equatio T rz X, T rz Y {0, } has a solutio, the = with e f X Y = Z c d g h k = I the proof of Theorem we use the followig lemma which ca easy prove by iductio: a Lemma If A M Q, A = b, b, d, f, h 0, the A = F a b e w f c w d F g h,, where w is a ratioal umber, F a b, F g h are polyomials of degree, a g a F F = b h b e w f Lemma for A M Z was proved i by K Bia lek ad A Grytczuk i [] The proof of Theorem Let the assumptios of the Theorem be satisfied From the equatio F we obtai Deote Hece, from 3 we have F Z X Z Y = I 3 A = Z X = XZ, B = Z Y = Y Z 5

3 Let A = From Lemma we obtai A = F a b e f w F c a b e d g f h c d w g h where w, w are ratioal umbers, A B = I 4 a, B = b c d e f g h a b, B = F e f w F c d w g h, 5 a g F F b F a b h g F h = = a e b f a e b f w, w By 4 ad 5 it follows that F a c b d w e g f w F h From 6 we give a b F e f w F a a F F = b b g g F F = h h c d w g h = c d w c d w = e f w e f w = 0 From the kow theorem of Cauchy we have det A = det B = From the other had from 5 we obtai deta = F a b F g h k 8 c e d f w, det B a g = F F c e w b h d f By 8, 9 ad the last equatio i 7 it follows that 6 9

4 From 5 we have From 7 we give From 0 ad 3 we obtai a F b By 5 ad 7 it follows that Similary we ca prove that a g a g F F = F F 0 b h b h T ra a g = F F b h T rb = F a b F g h g g F = F h F a b F a b h = F a b,, 3 4 g a F = F h b g a F = F h b a g F = F b h Therefore from 7, 6, 7, ad we obtai T ra a a = F F =, b b T rb g g = F F = h h Let λ, λ be the eigevalues of the matrix A The λ, λ are the eigevalues of the matrix A From 8 ad 8 we give T ra = λ λ =, 9 det A = λ λ = k 0 Let fλ = λ T raλ det A be the characteristic polyomial of the matrix A 7

5 The λ = T ra T ra 4 det A, λ = T ra T ra 4 det A are the characteristic roots of A We cosider the followig cases: 0 T ra = 0 The from we obtai λ = For = k from we obtai k, λ = k T ra = 0 what is cotrary with 9 For = k from 9 ad we obtai T ra = k =, thus what is cotrary for k, > 0 or k, < 0 k = 3 If k < 0 ad > 0 or < 0 ad k > 0, the k > 0 The the equatio 3 is true for = ad k = For = k, where k < 0 ad > 0 or < 0 ad k > 0 from we have det A = k k = Therefore 0 ad 9 are satisfied for = with k = 0 T ra = 8

6 The from we get λ = k, λ = k We have T ra = k k = k k 4 k k = k k 4 k k Let = k The we have k From 4, 5 ad 9 we obtai Assume that k < The k k k = k 5 k k k k = 6 k k k k > >, therefore the equatio 6 does ot hold Assume that k > The we remark that 6 is ot satisfied Let = k 9

7 The From 4, 7 ad 9 we have k If 0 < k <,the k k ad the equatio 8 is ot satisfied If k >,the k k = 0 7 k k k = 4 k 8 k k k > > 4 k k k k 4 k Similary as for the matrix A we obtai for the matrix B ad the proof of Theorem is fiished 0 Example Let X = Y = The Z = , Z = 0 Z X = XZ, Z Y = Y Z,, det X = det Y =, det Z =, We have X Y = Refereces det A = det XZ =, T ra = = Z [] Bia lek,k ad Grytczuk, A, 987, The equatio of Fermat i G k ad Q k, Acta Academiae Paedagogicae Agriesis Matematika,8/, 8-90 [] Cao, Z ad Grytczuk, A, 998, Fermat s type equatios i the set of x itegral matrices, Tsukuba J Math,,

8 [3] Cao, Z ad Grytczuk, A, 000, Some remarks o Fermat s equatio i the set of matrices, Acta Acad Paed Agriesis, Sectio Math, 7, [4] Domiaty, R Z, 966, Solutios of x 4 y 4 = z 4 i x itegral matrices, Amer Math Mothly, 73, 63 [5] Grytczuk, A, 997, Fermat s equatio i the set of matrices ad special fuctios, Studia Uiv Babes-Bolyai, Mathematica, 4, [6] Grytczuk, A, 998, O a cojecture about the equatio A mx A my = A mz, Acta Acad Paed Agrieusis, Sectio Math, 5, 6-70 [7] Grytczuk, A, 995, O Fermat s equatio i the set of itegral x matrices, Period Math Hugar, 30, 67-7 [8] Grytczuk, A ad Kurzyd lo, I,009, The ecessary ad sufficiet coditio for the solvability of Diophatie matrix equatio A mx A my A mz = A mw, Boleti de la Sociedad Matematica Mexicaa,5, 09-6 [9] Khazaov, A, 995, Fermat s equatio i matrices, Serodica Math J,, 9-40 [0] Le, M ad Li, C, 995, O Fermat s equatio i itegral x matrices, Period Math Hug, 3, 9- [] Patay, Z ad Szakacs, A, 00, O Fermat s problem i matrix rigs ad groups, Publ Math Debrece, 6/3-4, [] Qi, H, 996, Fermat s problem ad Goldbach problem over M Z, Liear Algebra App, 36, 3-35 [3] Ribeboim, P, 979, 3 Lectures o Fermat s Last Theorem New York: Spriger- Verlag [4] Vaserstei, N, 989, No-commutative Number Theory, Cotemp Math, 83,