Download Encyklopädie der Mathematischen Wissenschaften mit by Paul Bachmann (auth.), Wilhelm Franz Meyer (eds.) PDF

By Paul Bachmann (auth.), Wilhelm Franz Meyer (eds.)

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Extra resources for Encyklopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen: Arithmetik und Algebra

Example text

256; D. Seliwanoff, Moskau Math. Samml. 16, 1892, p. 469. 57) Uber solche rekurrente Reihen S. F. H. Siebeck, J. f. Math. 33, 1846, p. 71; D. Andre, Ann. ec. norm. (2) 7, 1878, p. 375; (2) 9, 1880, p. 209. = Comm. AI. 270; 1, p. 379. 9. Festst. einer Zahl ale Primzahl; Zerlegung grosser Zahlen in Faktoren. 577 zweitens, bei welchen a, b inkommensurabelj die Gleichung x 2 = x giebt Un = (1+ v'5)" - (1 2ny5 Jl5t, v: = n (1 + JI5)" + (12n +1 V'5r , und die Un sind die Zahlen von Fibonacci orier von Lame (Par.

Du! = 1 + 4 - zu betrachten 32). Man beschrankt sich auf die "eigentlichen" Auflosungen, in denen u ungerade ist. 1st (T", U") die kleinste Auflosung dieser Gleichungen, und ist T"2-DU"2 4 so erhalt man aus =+ 1, (T"+ ~"lCQr fiir k = 1, 2, 4, 5, 7, 8, . . alle eigentlichen positiven Auflosungen l ' h ung tt-Du 3 6 ,v,... (\ der Glmc 4 =1, und foour k =, aIIe Auf - IJu2 = 1. 1st aber losungen der Gleichung t 2 - T"! DU"! -4 = - 1, so erhalt man fiir k = 1, 5, 7, 11, 13, .. , aIle eigentlichen positiven 1, fiir k = 2, 4, 8, 10, ...

Diese Bedingung iet fiir die Darstellbarkeit notwendig, aber nicht hinreichend. Ersetzt man r und h durch andere gan~e Zahlen, filr welche ah - fJy = 1 iet, so bleibt der Kongruenzwert von n (mod. m) uncyll "gehOrif' zu diesem verandert: die Darstellung m = art ll 2brty Kongruemwerl von y'D (mod. m). AlIe Darstellungen von m, welche zu diesem Werle gehOren, bilden eine KIasee; aUe diese Darstellungen m = aa 2 2ba' y' cr'2 ergeben sich aus einer vermittelst der Transformationen der Form axll 2 bxy cyll in sich selbst oder such aus der Gleichung: + =+ + + + + + + + + aa' (b + y'D)r' = [art + (b + y'D)r] (t uVD), wenn rur t, u alIe Auflosungen der Pell'schen Gleichung gesetzt werden.

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