ON MAynEMATICAL TABLES. 325 



2) = c' + 2d' ; viz. decomposition of primes of tlie form 8ji+1 into the form 

 c--{-2cl-, A table extending from jj = 16 to 5943 (extracted from a JJH. table 

 calculated by Slnive) is given 



Jacobi, Crelle, t. xxx. (184G), ut sj/jjm, p. 180. 



This is carried by Eeuschle up to p = 12377, and for certain higher num- 

 bers up to 2-4889, as presently mentioned. 



40. Eeuschle's tables of the forms in question are contained in the work: — 



Reuschle, ' Mathematische Abhandlung &c.' (see anie No. 15), under 

 the heading " B. Tafeln zur Zerlegung der Primzahlen in Quadi'ate " 

 (pp. 22-41), They are as follows : — 



Table III. for the primes 6n + l. 



First part givesp=A-+3B' and 4j3 = L- + 27M-, from j) = 7 to 5743. 



Table gives A, B, L, M ; and those numbers which have 10 for a cubic 

 residue are distinguished by an asterisk. 



2) A B L M_ 



A specimen is 



37* 5 2 11 



viz. 37 = 5- + 3.2^ 148 = ir + 27.r; and asterisk shows that a-'= + 10 

 (mod. 37) is possible [in fact 34^=10 (mod. 37)]. 



Second part gives jj = A'+3B^ only, from j:) = 5749 to 13669. 



Table gives A, B and asterisk as before. 



Third part gives jj=A"+3B", but only for those values of p which have 

 10 for a cubic residue (viz. for which x^ ^ 10 (mod. p) is possible), from 

 jj = 13689 to 49999. 



Table gives A, B ; asterisk, as being unnecessary, is not inserted. 



Table IV. for the primes 4n-\-l in the form A'-+B", and for those which 

 arc also Sn-\-l in the form C-4-2D-. 



First part gives p = A" + B-, =C--|-2D'^, from^9 = 5 to 12377. 



Table gives A, B, C, D ; and those numbers which have 10 for a biquadratic 

 residue (.v^^lO (mod. j?) possible) are distinguished by an asterisk; those 

 which have also 10 for an octic residue (cc'^^10 (mod. p) possible) by a double 

 asterisk. 



... » A B C D 



A specimen is ^ 



Second part givesp=A- + B", from jj = 12401 to 24917 for all those values 

 of p which have 10 for a biquadratic residue (.i-'^ 10 (mod. 2?) possible). 



Table gives A, B ; and those values of j) which have 10 for an octic resi- 

 due, .r^^^lO (mod. j9) possible, are distinguished by an asterisk. 



Third part gives jj = C" + 2D", from jj = 12641 to 24889 for aU those values 

 of 2^ which have 10 for a biquadratic residue. 



Table gives C, D ; and those values ofp which have 10 as an octic residue 

 are distinguislied by an asterisk. 



41. A table by Zornow, Crelle, t. siv. 1835, pp. 279, 280 (belong to Me- 

 moir ' De Compositione numcrorum e Cubis integris positivis,' pp. 276-280), 

 shows for the numbers 1 to 3000 the least number of cubes into which each 

 of these numbers can be decomposed. Waring gave, without demonstration, 

 the theorem that every number can be expressed as the sum of at most 

 9 cubes. The present table seems to show that 23 is the only number for 

 which the number of cubes is =9 ( = 2. 2''' + 7 . 1'); that there are only 

 fourteen numbers for which the number of cubes is =8, the largest of these 



