Page:On the expression of a number in the form π‘Žπ‘₯Β²+𝑏𝑦²+𝑐𝑧²+𝑑𝑒².djvu/2

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12
Mr Ramanujan, On the expression of a number
  1. If , is an exception; and so
    (2Β·421).

    (2Β·43) .

    If , is an exception; and so

    (2Β·431).

    (2Β·44) .

    If , is an exception; and so

    (2Β·441).

    (2Β·45) .

    If , is an exception; and so

    (2Β·451).

    (2Β·46) .

    If , is an exception; and so

    (2Β·461).

    (2Β·47) .

    If , is an exception; and so

    (2Β·471).


    We have thus eliminated all possible sets of values of , , , , except the following 55:

    1, 1, 1, 1 1, 2, 3, 5 1, 2, 4, 8
    1, 1, 1, 2 1, 2, 4, 5 1, 2, 5, 8
    1, 1, 2, 2 1, 2, 5, 5 1, 1, 2, 9
    1, 2, 2, 2 1, 1, 1, 6 1, 2, 3, 9
    1, 1, 1, 3 1, 1, 2, 6 1, 2, 4, 9
    1, 1, 2, 3 1, 2, 2, 6 1, 2, 5, 9
    1, 2, 2, 3 1, 1, 3, 6 1, 1, 2, 10
    1, 1, 3, 3 1, 2, 3, 6 1, 2, 3, 10
    1, 2, 3, 3 1, 2, 4, 6 1, 2, 4, 10
    1, 1, 1, 4 1, 2, 5, 6 1, 2, 5, 10
    1, 1, 2, 4 1, 1, 1, 7 1, 1, 2, 11
    1, 2, 2, 4 1, 1, 2, 7 1, 2, 4, 11
    1, 1, 3, 4 1, 2, 2, 7 1, 1, 2, 12
    1, 2, 3, 4 1, 2, 3, 7 1, 2, 4, 12
    1, 2, 4, 4 1, 2, 4, 7 1, 1, 2, 13
    1, 1, 1, 5 1, 2, 5, 7 1, 2, 4, 13
    1, 1, 2, 5 1, 1, 2, 8 1, 1, 2, 14
    1, 2, 2, 5 1, 2, 3, 8 1, 2, 4, 14
    1, 1, 3, 5