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# On Rational Solutions of x3 + y3 +z3 = R.

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The earliest proof that every rational number (R) can be expressed as a sum of cubes of three rational numbers (x, y, z), not necessarily positive, was published in 1825 by S. Ryley, a schoolmaster of Leeds: formulae were given for x, y, z in terms of a parameter, such that every value of the parameter led to a system of values of x, y, z satisfying the above relation, and every rational value of the parameter led to a system of rational values of x, y, z. The later solutions referred to by Dickson are found to give the same results as Ryley's formula, as does another method, quoted in a modified form by Landau from a paper by the present writer. Thus it might almost be believed that Ryley's century-old result embodies all that is known with regard to the resolution of a number into three cubes, and that his formula is unique. I propose to examine the rationale of his method and the causes of its success; it will then appear that an infinity of similar formulae exist, and that one of them is at least as simple as his. It is convenient to state Ryley's formula, and the modification made by Landau, in section 2; and to generalize the method in section 3.

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page 92 note 1 Dickson, L. E., History of the Theory of Numbers, Vol. II, p. 726. Some further information which I have discovered is to be found in the Messenger of Mathematics, 51 (1922), 172.

page 92 note 2 Vorlesungen über Zahlentheorie, Band III, p. 216.

page 92 note 3 Proceedings of the London Mathematical Society (2) 21 (1922), 404409.

page 95 note 1 It does not seem possible to maks a more precise statement. Certain functions having irrational coefficients can be transformed into functions having rational coefficients, and would therefore serve our purpose. But it is also to be noted that, although coordinates of points of a cubic surface are generally expressible by rational algebraical functions of two parameters, yet in a bipartite cubic surface real points do not correspond to real parameters. It is not impossible that this irregularity in a matter of real and unreal numbers should have a parallel in a matter of rational and irrational numbers. Hence only a guarded statement can be made.

page 96 note 1 The functions X, Y, Z cannot be of lower order than three. It is supposed here that they are of order three, but they might be of higher order.

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• ISSN: 0013-0915
• EISSN: 1464-3839
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