THEORY OF COMMENSURABLES. 727 



prime number p, we can always find two numbers k and I each less than |/3 also 

 divisible by /?. 



In regard to k and I being odd or even, there are only three possible combina- 

 tions; one may be odd and the other even, in which case k'^ + P is of the form 

 4'i^ + 1 ; they may be both odd, in which case K^ + V' is double of an odd number 

 which odd number is the sum of an odd and even square, and still divisible by ^ ; 

 lastly they may be both even, in which case we can halve them, and continue to 

 halve, until one of the quotients be even and the other odd, without affecting the 

 divisibility of the sum of their squares by /?. 



Therefore, universally, if the prime number |5 be a divisor of s^ + f^ s and t 

 being prime to each other, it must also be the divisor of k'^ + /^ in which k and I 

 are each less than ^/3, the one being even and the other odd. 



Now k^ + 1^ being of the form 4w+ 1, cannot be divisible by any number ^ of 

 the form in— I, unless the quotient be also of the form 4w— 1 ; but k^ + P is less 

 than ^/5^ wherefore that quotient must be less than ^(3. And thus if any prime 

 number f3 of the form in— I can be a divisor of the sum of an odd and an even 

 square, some number less than its half, and consequently some prime number 

 less than its half, and of the same form 4n—\, must also be a divisor. 



If it were possible, then, that a prime such as 103 could divide the sum of an 

 odd and an even square, it would follow that some other prime of the same form, 

 and less than 51, would also be a divisor. The greatest prime of the form in— I, 

 under 51 is 47 ; if it, or any prime of the same form less than it, were a divisor 

 of k'^ + P; it would follow that some other prime (23 or under; would also be a 

 divisor. In this way we must ultimately arrive at the smallest prime of this 

 form, which is 3. Now there is no even number less than the half of 3, so that 

 3 cannot be a divisor of the sum of an odd and even square, and therefore we 

 conclude that no prime number, nor any other number, of the form 4^—1 can be 

 the divisor of the sum of two squares prime to each other. 



It is obvious that, whatever (3 may be, it is always a divisor of (l3sY + {^tf, 

 but then it is also the common divisor of /3s and ^t. 



17 Every prime number of the form in + l is the hypotenuse of one sole 

 right angled trigon in integer numbers. 



Every prime number of the form 4% + 1 can be resolved into two square num- 

 bers, one of which is odd and one even, but only into one pair. Let, then 



and 



wherefore 7 is the hypotenuse of a right-angled trigon, of which ip^—q^ and ipq 

 are the two sides. 



Here it may be observed that one of the sides is always divisible by 4. 



18. Every product of two prime numbers 7, 8, each of the form in + l is the 



