312 



KEPORT — 1875. 



A specimen is 

 a a — \ \ Cl- 



io 



11 



■1 I a'-l 



y.37 



13 



-1 la'-l ia"-li a^-1 «■=-! 



101 |41.27ll 9091 173.1371 9901 



(b. p. 21.) Specific prime factors for the numbers 2, 3, 5, 6, 7, 10 (the 

 powers 4, 8, 9 being omitted as coming under 2 and 3) for the exponents 1 

 to 42. 



A specimen is 



n 



19 



2"-l 



524287 



3»-l 



5«-l 



1597 . 363889 191 . x 



6"-l 



191 . X 419 . X 



7»_1 10"- 1 



where the x denotes that the other factor is not known to be prime. And so 

 where no number is given, as in 10^" — 1, it is not known whether the num- 

 ber ( = l + 10^ + 10^ . . +10"*) is or is not prime. 



Addition, p. 22. For «=2, the complete decomposition of the prime factor 

 of 2" — 1 is givenfor values of n, =44,45 ... at intervals to 156. 

 A specimen is n / 



44 397 . 2113, 

 viz. 2="-2'^ + 2'^..-2= + l, =838.861=397.2113. 



n=31, Format's prime. h = 37, the first case for which the de- 

 composition is not given completely. «=41, the first case for 

 which no factor is known. 



16. Gauss, Tafel zur Cyclotechnie, Werke, t. ii. pp. 478-495, shows for 

 2452 numbers of the several forms a" + l, «"+4, «- + 9, .... rt- + 81, the 

 values of a such that the number in question is a product of prime factors 

 no one of which exceeds 200, and exhibits all the odd prime factors of each 

 such number. The tabic is in nine parts, zerlegbare «^ + l, zcrlcgbare o"+4, 

 &c., with to each part a subsidiary table, as presently mentioned. Thus a 

 specimen is 



zerlegbare rt- + 9. 



5 . 13 . 17 . 17 . 89 . 113 . 157 . 173 . 197 . 197. 



1- + 9, odd prime factor is 5, 

 2= + 9, „ „ 13, 



4" + 9, „ factors are 5, 5, 

 and so on. 



And the subsidiary table is 



5 

 13 

 17 



1, 4,79 



2, 11, 41 



5, 29. 46, 379, 1042 



showing that the numbers a for which the largest factor is 5 are 1, 4, 79; 

 those for which it is 13 are 2, 11, 41 j and so on. 



