TALBOT6375

26 Lincoln’s Inn Fields

12 October 1847

Dear Sir,

I shall feel much flattered and pleased to receive your kindly promised specimens of the Talbotype process. Your remarks on my recent and continuing labours are encouraging as they afford evidence of some of that scientific sympathy of which there is so unfortunate a lack –, at least very little of it has ever fallen in my way – here in England. I think with you that very much remains to be done in the Theory of Numbers and hope to be of some use in simplifying some of the most essential parts of the Doctrine, so far as it is as yet developed, as I find them handled in Legendre who I believe derives chiefly from Lagrange and in Gauss. Barlow’s supposed Discovery is a mere blunder – you were quite right in your impression of his having announced a Complete demonstration of the Insolubility of x^m + y^m + z^m = 0 in his book on the properties of numbers. Besides the cases you name, this had been lately proved (I believe I was informed in Paris) quite recently by Lamé ¹ for m = 7.

He also announced the general demonstration as you will find in the Cptes Rendus; but this was a fallacy as he himself I believe is now ready to admit.

I have confined myself for the present to the case of the Cubic equation which takes in x³ + y³ +z³ = 0 f as a mere and very limited particular instance I shall give in Taylors next number the conditions under which we can recognise the Insolubility of the equations

a(x³ + y³ + z³) + c(x²(y + z) + y²(z + x) + z²(x + y)) ² + exyz = 0

This may always be made to depend on the an equation of the form x³ + y³ + Az³ = Mxyz

I have likewise discovered the Truth of the presentiment darkly shadowed out at the conclusion of my last paper in the Magazine.

One Whenever 27A − M³ contains no positive prime factors of the form 6n + 1 and it satisfies certain conditions of form the number of <illegible deletion> Primitive Solutions is always limited and it is easy to assign a superior limit to the number of such solutions. Very frequently it can be shown that there is but one such system for instance this is always the case when M = 0 and A is any power of a prime number not of the form 6n + 1; of course if this power is a cube the equation is insoluble; but if any other power then there exists one solution, i.e. one system of values of x, y, z from which the values of x, y, z in all other systems <illegible deletion> satisfying the equation x³ + y³ + Az³ = 0 can be determined reduced. For instance if we take the equation x³ + y³ + 9z³ = 0 By means of the obvious system x = 1 y = 2 z = −1 I can find every all the possible values simultaneous values of x, y, z which will satisfy the above equation. It is strange that so remarkable a fact should never have been suspected before.

I remain My dear Sir,

Yours very Truly

J J Sylvester

Envelope:

H. Fox-Talbot Esq^r
Athenæum Club