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



erroneous, viz.: — " Cor. 2. In the same manner it may he shown, if there be 

 several fractions, and one of them be in its lowest terms, and contain a factor in 

 its denominator, that is not common to all the other deiwminators, the sum of these 

 fractions cannot be an integer." As Baelow's demonstration of Fermat's theorem 

 reposes on this Corollary, that demonstration falls to the ground, and a true 

 demonstration of the theorem still remains to be sought for. 



II. There is a well known and very remarkable theorem, that " Every prime 

 number of the form An + \ is the sum of two squares, and in one way only." 



The most simple proof of this appears to consist in the following series of 

 propositions : — 



(1.) The product of the sum of two squares by a similar quantity is likewise 

 the sum of two squares, and in two ways, — 

 Because {a" + W) (c^ + d^) = {ac + hdf + {ad- hcf ; 



and also, = {ac — bdf + {ad + bey 



(2.) The sum of two squares can only be divided by a quantity of like form. 

 (3.) By Wilson's theorem, a prime j9 always divides 1, 2, 3, . . . . (7^- 1) + 1, 

 and this product may be written 



l.(i,-l) . . . 2.0,-2) . . . 3. (i>-3) . . . &c. 

 or, {p-l) (2p-22) (3p-32) . . . &c. 



or omitting the multiples of jf, and observing that the number of factors is even, 

 if p is of the form 4n+l, the product may be written 



Px22x32 . . . {2nf = [l .2.3... 2nf = QK 



Therefore p divides Q^ + 1 the sum of two squares. Therefore 2^ is itself the sum 

 of two squares. 



Barlow, at p. 205 of his work, gives the converse of this theorem, and says, 

 that a number of the form 4n + 1 is necessarily a prime number, if it is the sum 

 of two squares, in one way only. Suppose, however, that we take for example 

 the number 45. This is the sum of two squares 3G + 9, and in one way only. 

 Nevertheless, the number 45 is not a prime, as it ought to be by this rule. This 

 shows how much caution is necessary in writing on this branch of mathematics. 

 The fact is, the theorem only holds good in case the two squares 3lvq prime to each 

 other. Now, 36 and 9 are not so ; and, consequently, the conclusion that their 

 sum is a prime number is erroneous. With this limitation, however, I believe 

 the theorem is correct. There is one apparent exception, however, which should 

 be pointed out. The square of a prime number of the form 4n + 1 is of the same 

 form, and is the sum of two squares in one way only. Thus, b^, or 25=16 + 9 

 and 13^ or 169 = 144 + 25. The test, therefore, appears to fail in these instances. 

 But in fact it holds good ; for 25 is not only the sum of the squares 16+9, but 

 also of 25 + 0, and this consideration applies to all similar cases. 



