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XXXIII. An account of a Discovery in the Theory of Numbers 

 relative to the Equation A.x^ + Bj/^ + Cs^ = Tixyz. By J. J. 

 Sylvester, Esq-^ M.A., F.R.S.^ 



First General Theorem of Transformation. 

 TF in the equation Kx^-\-^y^->tC^ = 'Dxyz . . (1.) 



A and B are equal, or in the ratio of two cube numbers to 

 one another, and if 27 ABC — D^ (which I shall call the De- 

 terminant) is free from all single or square prime positive 

 factors of the form 6n+ 1, but without exclusion of C7ibic factors 

 of such form, and if A and B are each odd, and C the double 

 or quadruple of an odd number, or if A and B are each even 

 and C odd, then, I say, the given equation may be made to 

 depend upon another of the form 



A'm3 + Wv" + C'«)3 = B'xyz ; 

 where A'B'C' = ABC 



D'=D 

 u.v .iv =some factor of z. 

 The following are some of the consequences which I deduce 

 from the above theorem. In stating them it will be convenient 

 to use the term Pure Factorial to designate any number into 

 the composition of which no single or square prime positive 

 factor of the form 6 n+1 enters. 



The equations x^ + 2^ + 22^=Dxys: 

 a^ + y^ + 4!Z^ = T)xyz 

 2a^ + ly^ + z^ = Yixyz 

 are insoluble in integer numbers, provided that the Determi- 

 nant in each case is a Pure Factorial. 



The equation a^-\-y^-\- K^'rs.^Vtxyz 

 is insoluble in integer numbers, provided that the Determinant, 

 for which in this case we may substitute A — 27B^, is a pure 

 factorial whenever A is of the form 9w+l, and equal to 

 2p3!±i Qp 4.p3i±i^ jp being any prime number whatever. 



I wish however to limit my assertion as to the insolubility 

 of the equations above given. The theorem from which this 

 conclusion is deduced does not preclude the possibility of two 

 of the three quantities a?, ?/, z being taken positive or negative 

 units^ either in the given equation itself or in one or the other 

 of those into which it may admit of being transformed. Should 

 such values of two of the variables afford a particular solution, 

 then instead of affirming that the equations are insoluble, I 

 should affirm that the general solution can be obtained by 

 equations in finite differences f. 



* Communicated by the Author. 



t Take for instance the equation ifi-\-y^-\-^z^'=9xyz. The Determinant 



