726 



MR EDWARD SANG ON THE 



between and 0, the one half is the converse of the other : farther, no two re- 

 mainders in the half group are alike. 



The import of these assertions may be seen by taking any prime number, as 

 11, and dividing the successive square numbers 0, 1, 4, 9, 16, 25, 36, 49, &c. by 

 it. The remainders are found to be 0, 1, 4, 9, 5, 3, 3, 5, 9, 4, 1, 0, 1, 4, 9, 5, 3, 8, 

 5, 9, 4, 1, 0, and so on. 



In reference to the divisor «, the natural numbers 0, 1, 2, 3, &c., belong to 

 the forms 



na + 

 na + 1 

 na + 2 



na + 

 na + 

 na- 

 na— 



a-S 



2 

 g-l 



2 

 a-1 



/ na + 

 na + 1 

 na + 4: 



2 

 a-S 



o 



na — 2 

 na — 1 

 na — 



of which the 

 squares are 

 of the same 

 forms with 



na + 



a2-6a + 9 



na + 



a^ — 'Ia + l 



I 



na 



na + 



na + i: 

 na + 1 

 na + 



a^-2a+l 

 4 



'»2_6a + 9 



From which it is obvious that the order of the remainders in the latter half of 

 the group is inverse of the order in the former half 



Also the same remainder cannot occur twice in the half group. For if any 

 two squares, as k"^ and /", k and / being each less than |a, had the same remainder, 

 their difference k'^— P would be divisible by a. Now both k + l and k—l, the fac- 

 tors of k^ — P, are less than a, and their product cannot possibly be divisible by a. 



16. The sum of two squares which are prime to each other is not divisible by 

 any number of the form 4n—l. 



Every composite number which is of the form in— I, must have an odd 

 number of factors of the same form combined with some or no factors of the 

 form 4:71 + 1. It will be enough to show that the sum of two squares prime to 

 each other cannot be divided by any prime number of the form 4w — 1. 



If the two numbers s and t be both odd, s^ + f is even, and its half is the sum 

 of two squares and also odd ; and if / 4- f be divisible by any prime (3, its half 

 must also be divisible by /3. Wherefore we have only to show that the sum of 

 an odd and an even square can never be divisible by jS v?=^ 4w — 1. 



Let us suppose that s=p^'+ik, t=q^^l, then 



s2 + i^ = (p^ + g^) /32d=2 (pk + ql) + Ic" + V 



k and I being each less than ^^. Wherefore if the sum s^ + f be divisible by a 



