Download 250 problems in elementary number theory by Waclaw Sierpinski PDF

By Waclaw Sierpinski

ISBN-10: 0444000712

ISBN-13: 9780444000712

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Example text

H. durch u 2 teilbar. Oder so: In der Determinante E ±Pa qb Y c haben alle sechs Glieder einen und denselben Nenner j(a) j(b) j(c), welcher in u aufgeht. Mit u multipliziert, werden also diese und alle ähnlichen Determinanten zu ganzen Funktionen. Setzen wir auch das sekundäre System aus höheren Gleichungen zusammen und geben ihm, wie vorhin, lauter Differentialkoeffiziepten desselben Ranges oc zu Elementen, so dürfen wir auch da keine höhere Teilbarkeit erwarten. Das sekundäre System wird folgende Form bekommen: u (~ 1(1) (PI ~ + ql V + ..

E (a p - h i) + E {i P(a . D d f g f> = E {a h (a . In 0 - + a h (P . 0 j k m - i + i P (p . bc e -- i . d f g)}. h . D bc e) h . j k m) + 2 E a phi (a p . 0 In o)}, h i) 56 Abhandlung Nr. 24 Hierbei ist zu bemerken, daß D 2 bce=2djkm, D 2 dfg=2dlno, Ddjkm=3jkm, Ddlno=3lno ist. j - 2 Qj - 3~) = D (1; + l) - 3 (f). Der Ausdruck genügt also allen Bedingungsgleichungen von der Form D = 0 und ist daher eine vollständige Hyperdeterminante. , (M, N) = m~O gesetzt wurde, wenn M, N homogene Funktionen n-ten Grades der Variablen a, b, c, d, e, f, g, h bezeichnen.

Wir haben LI = = 1 2" (q, r, s) = 1 2" (a d - b c, a h + d e - b g - c t, eh - t g) ! Ha d, a h, eh) - (a d, a h, t g) + (a d, d e, eh) - (a d, d e, t g) - (a d, c t, e 11) - (b c, a h, eh) + (a d, b g, t g) + (b c, a h, t g) - (b c, d e, eh) + (a d, c t, t g) + (b c, d e, t g) + (b c, b g, eh) - (b c, b g, t g) + (b c, c t, eh) - (b c, c t, t g)}. - (a d, b g, eh) Dieses Aggregat zerfällt in drei Gruppen, welche durch die Glieder 1 -2" (a lz, ad, eh), (I) - 2" (a h, b c, t g), (Il) - 2" (a h, a d, t g) (lU) 1 1 repräsentiert sind.

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