584 MR TALBOT ON SOME MATHEMATICAL RESEARCHES. 



It is remarkable that this theorem should not be found, as I think, in the 

 treatises of Algebra, for it furnishes a negative test as to whether the roots of a 

 given equation, x 3 + qx — r = 0, are whole numbers. If, for any reason, it is sup- 

 posed that they are so, let the coefficient q (taken positively) be divided by 3 ; 

 then, if it leaves the remainder 2, the roots cannot be whole numbers. 



Hence, if the roots of the equation are whole numbers, the coefficient q cannot 

 be taken ad libitum. Indeed, by substituting various numbers for x and y in the 

 formula q — x 2 + xy + y 2 , it will be seen that, of the numbers below 100, only 27 

 can be values of q. These numbers are 3, 7, 12, 13, 19, 21, 27, 28, 31, 37, 39, 43, 

 48, 49, 52, 57, 61, 63, 67, 73, 75, 76, 79, 84, 91, 93, 97. It will be seen that all 

 these numbers are of the form 3n or 3n + 1, never of 3n + 2. Of all these values 

 of q only the number 91 occurs twice, namely, when we suppose x = l, y=9, or 

 else x — 5, y — 6. From whence we may infer, that if the roots of x 3 + qx — r =0 

 are whole numbers, they may, in most cases, be determined from the value of q 

 alone ; but that if q has more than one value, r has the same number of values. 



It is also observable that this list contains every prime number (of form 3w+ 1) 

 below 100. Does this law continue ? and may we infer that integer values of x 

 and y always exist, which will satisfy x 2 + xy + y 2 = q, q being any prime of 

 form Zn + 1 ? 



N.B. — Since this was written, I have tried the second century of numbers, or 

 those between 100 and 199, including the latter. I find that only 28 of these can 

 be values of q or x 2 +xy+y 2 , and of these only two occur twice, namely 133 and 

 147. And I find that every prime of the form 2>n + 1 is found in the list ; there- 

 fore the induction holds good so far. There is a well-known theorem " that every 

 prime number of the form in + 1 is the sum of two squares, and in one way 

 only." Perhaps it is true (which I only offer as a conjecture) that " every 

 prime number of the form 3n + l is of the form x 2 + xy+y 2 , and that in one 

 way only." 



We have seen that the equation — q = x 2 + xy+y 2 , expresses the relation between 

 any two roots x and y. The solution of this equation gives 2y + x=V—4q — 3x 2 . 

 But 2y + x=y — z, because x +y + z = 0. Whence we derive this theorem: If .z 

 be any root of the equation x 3 + qx — r=0, the difference of the other two roots 



= V — 4q — Sx 2 , which may be called R. Since, then, y + z = — x and y — z= R, 



R — x — R — x 



we obtain y — — ^ — > z — — 2 — * ^^ s seems much easier than the common 



method, which prescribes (when one root a is known) that we should divide the 

 equation by x — a, and solve the resulting quadratic. Again, since 2y + x=y — e 

 is a whole number, V — 4q — Sx 2 must be a whole number : from which the 

 important consequence follows, that in any equation x 3 + qx — r=0 whose roots 

 are integers, — 4q — 3x 2 is necessarily a square, whichever of the roots is taken 

 for x. 



