The Pell Equation

The Pell Equation

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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1912 edition. Excerpt: ...the problem of the Pell equation by his method of substitutions. He has avoided the use of continued fractions and has shown that if we form by the method he indicates the period of a quadratic form of determinant A we may infer the complete solution of the equations x2--Ay2 = 1 and x2--Ay2 = 4 from the automorphics of any reduced form, according as the form is properly or improperly primitive.2 The following theorem from the Disquisitiones Arithmeticae3 shows how he connected the Pell equation with the theory of quadratic forms. 1 J. L. Lagrange, "Euler's algebra with additions by M. de la Grange," translated by Hewlett, Chap. VIII, p. 578, London, 1840. On this page, through a misunderstanding of Ozanam, Lagrange seems to think that the solution of Lord Brouncker should be credited to Fermat. C. F. Gauss, "Disquisitiones arithmeticae," 198-202, p. 259-273, Leipzig, 1801. Op. cit., 162. "Werke," vol. I, p. 129, 2d ed., Gottingen, 1870. If the form AX2 + 2BXY + CY2 = F implies the form ax2 + 26x2/ + cy2--f, and if a certain transformation is given which transforms the first into the second; from this to deduce all the transformations which produce the same result. The solution is substantially as follows: If the given transformation is X = ax + @y and Y = yx + by, we first let another similar transformation produce the same result. Let this new transformation be X = a'x + fi'y, Y = y'x + S'y. Let the determinants of the forms F and f be D and d, and let ad-@y = e, a'i'-$'y' = e'. Then d = De2 = De'2, and since by hypothesis e and e' have the same sign, it follows that e--e'. We have moreover the following six equations: whence a'c'-D(a8'-700037'-5a') = 62, and by subtracting D(a5-07)(a'5'-0V) more

Product details

  • Paperback | 38 pages
  • 188.98 x 246.13 x 2.03mm | 90.72g
  • Miami Fl, United States
  • English
  • black & white illustrations
  • 1236633598
  • 9781236633590