Page:On the expression of a number in the form 𝑎𝑥²+𝑏𝑦²+𝑐𝑧²+𝑑𝑢².djvu/11

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in the form
  1. (8·3) .

    If , take . Then

    is of one of the forms

    .

    If , take . Then

    is of the form . In either of these cases is of the form .

    This completes the proof that there is only a finite number of exceptions. In order to determine what they are in this case, we have to consider the values of , between and , for which and

    .

    But the numbers which are multiples of and which cannot be expressed in the form are the numbers

    .

    The exceptional values of required are therefore those of the numbers

    (8·31)
    which lie between and and are of the form

    (8·32).
    But in order that (8·31) may be of the form (8·32), must be if is of the form , and may have any of the values if is of the form . It follows that the only numbers greater than which cannot be expressed in the form (7·1), in this case, are the numbers of the form

    ,

    lying between and , where if is of the form , and if is of the form .

    This completes the proof of the results stated in section 7.