Higher Arithmetic : An Introduction to the Theory of Numbers (9781139734653) by Davenport H

Author:Davenport, H.
Language: eng
Format: epub
Publisher: Cambridge Univ Pr

For if h is any solution of the congruence, and (n, h, l) the corresponding form, we can apply to this form the special substitution

x = X + uY, y = Y,

where u is any integer. This gives the form

n( X + uY )2 + h( X + uY )Y + lY2.

The first coefficient is still n, and the middle coefficient, instead of being h, is now h + 2un. Consequently two forms with first coefficient n and with middle coefficients which differ by a multiple of 2n are necessarily equivalent. So it is enough to consider those forms for which h satisfies the inequality (9) as well as the congruence (8).

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