MR H. F. TALBOT ON THE THEORY OF NUMBERS. 51 



Although a prime of the form 4:n+l, is always the sum of 2 squares, yet a rule 

 is wanting to determine these squares. The following answers for one case: — 

 Let p be the prime. Try if 2p — l, is a square, and if so, call it g^. 



Then p={'-ry-{'i'y 



Emmjjle.—Let p = l86l .-. 2p-l = 3721 = 6P=/ .-.^=302 + 312. 



§ 3. Remarks on Barlow's Theory of Numbers. 

 Peter Barlow, of the Royal Military Academy, a mathematician of eminence, 

 and author of a volume of tables most useful to all persons engaged in numerical 

 computations, and believed to be exceedingly accurate, published in 1811 a work 

 entitled " An Elementary Investigation of the Theory of Numbers." This book, 

 which gives much useful information on a subject at that time little known to the 

 English reader, contains a few errors which ought to be pointed out, lest they 

 should acquire credit, by having appeared in a work of authority. 



I. It is well known that mathematicians have never been able to find the 

 demonstration of Fermat's theorem, which asserts that a" + h"—c'', is an impossible 

 equation, if n is an integer number greater than 2. Nevertheless, Barlow, at 

 p. 160 of his work, professes to give a demonstration of this theorem. Subsequent 

 mathematicians, however, have tacitly ignored Barlow's demonstration, and the 

 question has continued to be proposed from time to time by the French Institute 

 and other learned societies, without receiving any solution. It is worth while, 

 therefore, to inquire for what reason Barlow's demonstration has been put aside. 

 Before treating of the general problem, to satisfy the equation «" + &"=c", he treats 

 of the particular case a^ + 1f = c% and as he treats this exactly in the same way 

 pp. 132-140, one explanation will suffice for all. It appears to me that the error 



of the demonstration lies in p. 139, where he obtains an equation r = 6 ^ 



X ^ r tr st 



and saySj^rs^, that because r, s, t, are prime to each other, each of the above 

 fractions is in its simplest form ; and, secondly, that they each contain a factor in 

 their denominator, that is not common with the other denominators ; and there- 

 fore, these fractions cannot, anyhow combined, be equal to an integer, by Corol- 

 lary 2 of Art. 13. But this theorem is not true. Take for example the equation 



7 8 3 



2:0 + 3:5+2^ = 8. According to the theorem, S cannot be an integer, because the 



fractions are in their lowest terms, and each denominator contains a factor, that 

 is not common to the other denominators. 



But on trial, we find that S=2, an integer. Turning, therefore, to the Corol- 

 lary mentioned, which is found at p. 20, we see that it rests upon a theorem 

 in p. 19, viz. : — " The sum of two fractions in their lowest terms, of which the 

 denominator of the one contains a factor not common with the other, cannot be 

 an integer." This may be admitted ; but Cor. 2, which follows, appears to be 



