334 KEPORT — 1875. 



The foregoing synopsis of EeuscUe's tables in the ' Berliner Monatsberichte' 

 "was written previous to the publication of Eenschle's far more extensive 

 work. It is allowed to remain, but some explanations which were given 

 have been struck out, and were instead given in reference to the larger work. 



Reuschle, " Tafeln complexer Primzahlen welche aus Wurzeln der Einheit 

 gebildet sind." Berlin, 4° (1875), pp. iii-vi and 1-667. 



This work (the mass of calculation is perfectly wonderful) relates to the 

 roots of unity, the degree being any prime or composite number, as presently 

 mentioned, having all the values up to and a few exceeding 100 ; viz. the 

 work is in five divisions, relating to the cases 



I. (pp. 1-171), degree any odd prime of the first 100, viz. 3, 5, 7, 11, 

 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 

 83, 89, 97 ; 

 II. (pp. 173-192), degree the power of an odd prime 9, 25, 27, 49, 81 ; 



III. (pp. 193-440), degree a product of two or more odd primes or their 



powers, viz. 15, 21, 33, 35, 39, 45, 51, 55, 57, 63, 65, 69, 75, 77, 

 85,87,91,93, 95, 99, 105; 



IV. (pp. 441-466), degree an even power of 2, viz. 4, 8, 16, 32, 64, 128 ; 

 V. (pp. 467-671), degree divisible by 4, viz. 12, 20, 24, 28, 36, 40, 44, 



48, 52, 56, 60, 68, 72, 76, 80, 84, 88, 92, 96, 100, 120 ; 



the only excluded degrees being those which are the double of an odd prime, 

 these, in fact, coming under the case where the degree is the odd prime 

 itself. 



It would be somewhat long to explain the specialities which belong to 

 degrees of the forms II., III., IV., V. ; and what foUows refers only to 

 Division I., degree an odd prime. 



For instance, X = 7, \ — 1=2*3, the factors of 6 are 6, 3, 2, 1; and there 

 are accordingly four divisions, 



I. a a prime seventh root or root of a" + a" + a^ + a'^ + a + 1 = 0. 



, a root of ,.+ ,--2,-I = 0| W;Zj. + 2.; ^^ 



III. r)„=a + a+a\ r]^ = a^ + a' + a\ 01 rj a root of »,- + j, + 2 = 0. 



IV. Real numbers, 



I. p = 7m + l. (1) gives for the several prime numbers of this form 29, 43 

 , . . . 967 the congruence roots, mod. 2> ; for instance, 



p a a^ a^ a* a' a* 



29 - 5 -4 -9 -13 +7 -6 

 43 4-11 _8 -2 +21 +16 +4 



this means a^— 5 (mod. 29) ; then a'^25, ^—4, a^^20,^ — 9, (fee, values 

 which satisfy the congruence a° + a° + a' + a^ + a" + a+1^0 (mod. 29). 



(2) gives under the simple and the primary form the prime factors /(a) 

 of these same numbers 29, 43 ... . 967 ; for instance, 



2) f(a) simple. /(a) primary. 



29 a + a=-a' 2 + 3a -a- + 5a' -2o* + 4a'' 



43 a'-\-2a' 2a-2a- + 4a*-a'-5a\ 



The definition of a primary form is a form for which /(a) /(a~*) ^f(iy 

 mod. X, and/(a)^/(l) mod. (1 — a)". The simple forms are also chosen so 

 as to satisfy this laxt condition ; thus f(a) = a + a'—a^, then /(I)— /(a) = 

 1 - a - u- + cr = (1 - tO'(] + «), = mod. (1 - ay. 



