324! REPORT — ISrS. 



It is clear that the series of tables might be coutimied iudefiQitely, viz. 

 there might be a table IV. giving the developments of 



as C3 



S', :; , ;, and so on. 



1 — .r 1 — .T.l — ,^' 



An interesting table would be one composed of the first lines of the above 

 series, viz. a table giving in its successive lines the developments of S, S^ B', 

 S', &c. 



There are throughout the work a large number of numerical results given 

 in a quasi-tabular form ; but the collection of these, with independent expla- 

 nations of the significations of the tabulated numbers, would be a task of 

 considerable labour. 



Art. 5. [F. 15. Quadratic fo7-ms a^-f b^ ^c, and Partitions of Numbers into 

 squares, cubes, and biquadrates.'] 



37. The forms here referred to present themselves in the various complex 

 theories, thus N = «' + &% —(a-\-bi)(a — bi); this means that in the theory 

 of the complex numbers a + bi (a and b integers) N is not a prime, but a 

 composite number. It is well known that an ordinary prime number ^ 3, 

 mod. 4, is not expressible as a sum cr + b'-, being, in fact, a prime in the 

 complex theory as well as iu the ordinary one, but that an ordinary prime 

 number ^ 1, mod. 4, is (in one way only) =rt--f-6-; so that it is in the 

 complex theory a composite number. A number whose prime factors are 

 each of them ^ 1, mod. 4, or which contains, if at all, an even number of 

 times any prime factor hs 3, mod. 4, can be expressed in a variety of ways 

 in the form a" + h^ ; but these are all easily deducible from the expressions 

 in the form in qiiestion of its several factors ^ 1, mod. 4, so that the re- 

 quired table is a table of the form p = a- + b', p an ordinary prime number 

 ^ 1, mod. 4 : a and b ai-e one of them odd, the other even ; and to render 

 the decomposition definite a is taken to be odd. 



p> — a'-{-b'; viz. decomposition of the primes of the form 4n-|-l into the 

 sum of two squares, a table extending from jj = 5 to 11981 (calculated by 

 Zornow) is given 



Jacobi, Crelle, t. xxx. (1846), pp. 174-176. 



This is carried by Reuschle, as presently mentioned, up to p> = 24917. 

 Reuschle notices that 2713 = 3" -|- 52" is omitted, also 0997 = 39" -|- 74', and 

 that 8609 should be =47"-}- 80". 



38. Similarly primes of the form Gn -\- 1 are expressible in the form 

 2)=a'^-\-3b'. (Observe that w being an imaginaiy cube root of unity, this 

 is connected with p>' = (« + bco) (a -\- bw^), = d^ — ab -\- b'\ viz. we have 

 4p' = {2a — by -\- 3b'' ; or the form cr -\- 3b' is connected with the theory of the 

 complex numbers composed .of the cube roots of unity.) 



2} — d--^3b'- ; viz. decomposition of the primes of the form Gn-\-l into the 

 form a- + 3b-. A table extending from p = 7 to 12007 (calculated also by 

 Zornow) ig given 



Jacobi, Crelle, t. xxx. (1846), id sui^ra, pp. 177-179. 



This is carried by Reuschle up to j}= 13369, and for certain higher num- 

 bers up 49999, as pi-esently mentioned. Eeuschle observes that 0427 = 80" 

 -)-3 . 3" is by accident omitted, and that 6481 should be =4r-f 3 . 40^ 



39. Again, primes of theform8)i-|-l are expressible in the form p=«" 4- 25" 

 (or say =c' + 2d'), the theory being connected with that of the complex 



'numbers composed with the Sth roots of unity (fourth root of - 1, = - .- -). 



