2850 Congruences other than Linear ; Cubic and Higher Residues 2850 



Primitive roots of primes, criteria. Zsigmondy, 

 K. Mh. Mth. Ps. 4 (1893) 79-. 



, properties. Jonquieres, de. 

 C. E. 122 (1896) 1451-. 



and their residues. Forsyth, A . R. 



Mess. Mth. 13 (1884) 169-. 

 , solution of question proposed by Fermat. 



Lipschitz, R. Bll. Sc. Mth. 22 (1898) 123-. 

 , theorem. Le Besgue, V. A. C. E. 64 



(1867) 1268-. 



, two theorems of Gauss generalised. 

 Arndt, F. Crelle J. 31 (1846) 326-. 



and secondary roots of primes. Jonquieres, 

 de. C. E. 123 (1896) 374. 



Pure circulates in various scales of notation. 



Reynolds, B. Mess. Mth. 28 (1899) 88-. 

 Quadratic, cubic and biquadratic residues. 



Pepin, T. Em. N. Line. At. 38 (1885) 197-. 



and cubic residues, theory. Meyer, G. 

 Arch. Mth. Ps. 63 (1879) 1-. 



Quintic, octavic, and duodecimic residues, com- 

 plex primes in theory of. Jacobi, C. G. J. 

 Crelle J. 19 (1839) 314-. 



residues. Pepin, . C. E. 76 (1873) 151-. 



and integers formed from 5th roots of 

 unity. Schivering, K. Z. Mth. Ps. 27 (1882) 

 102-. 



Eeciprocals of primes, to determine prime itself 



from number of figures in reciprocal. Shanks, 



W. E. S. P. 22 (1874) 381-. 

 , number of figures in period. Shanks, 



W. [1873-75] E. S. P. 22 (1874) 200-, 384-; 



23 (1875) 260-, 261. 

 , periods in. Shanks, W. [1872] Mess. 



Mth. 2 (1873) 41- ; 3 (1874) 52-. 

 , . Salmon, G. [1872] Mess. 



Mth. 2 (1873) 49-. 

 Eeciprocity, bicubic, between real and complex 



numbers. Gmeiner, J. A. Mh. Mth. Ps. 3 



(1892) 179-, 199-. 



law, bicubic. Gmeiner, J. A. Wien Ak. 

 Sb. 100 (1891) (Ab. 2a) 1330- ; 101 (1892) 

 (Ab. 2a) 562-. 



, biquadratic, theorem. Dintzl, E. Mh. 



Mth. Ps. 10 (1899) 88-. 

 , cubic. Eisenstein, G. Crelle J. 27 



(1844) 289-; 28 (1844) 28-. 

 , . Gegenbauer, L. Wien Ak. Sb. 81 



(1880) (Ab. 2) 436-. 



, , analytic proofs. Dantscher, V. 

 Mth. A. 12 (1877) 241-. 



, , theorem. Dintzl, E. Mh. Mth. Ps. 



10 (1899) 303-. 



laws, general, of power residues. Rummer, 

 E. E. Berl. B. (1850) 154- ; Berl. Mb. 



1858) 158- ; Crelle J. 44 (1852) 93-; 56 



1859) 270- ; Berl. Ab. (1859) (Mth.) 19-; 

 1861) (Mth.) 81-. 



, higher, and laws connected with them. 



Eisenstein, G. Crelle J. 39 (1850) 351-. 

 , of power residues. Pepin, T. [1877] 



C. E. 84 (1877) 762- ; Em. N. Line. At. 31 



(1878) 40-. 

 Eesidue of a binomial coefficient divisible by 



p n , with respect to (p n+l ). Glaisher, J. W.L. 



QJ. Mth. 30 (1899) 349-. 

 with respect to a prime modulus. 



Glaisher, J. W. L. QJ. Mth. 30 (1899) 150- . 



Eesidue, two as a 16-ic. Cunningham, (Lt.- 

 Col.) A. [1895] L. Mth. S. P. 27 (1896) 85-. 



Eesidues. Lebesgue, V. A. C. E. 3 (1836) 

 439-. 



. Arndt, F. Crelle J. 31 (1846) 333-. 



of binomial coefficients with respect to p 3 . 

 Glaisher, J. W. L. QJ. Mth. 31 (1900) 

 110-. 



in geometric progressions. Prouhet, E. 

 N. A. Mth. 5 (1846) 175-, 652-, 675-. 



of higher orders. Gazzaniga, P. Ven. I. 

 At. (1885-86) 1271-. 



index-sums for primes of form p = 2 ^ + 1, 



symmetrical and complementary distribution. 

 Hermes, J. Arch. Mth. Ps. 4 (1886) 207-. 



(mod. k) , of symmetric functions. Mitchell, 

 0. H. Am. J. Mth. 4 (1881) 25-. 



and primitive roots, theorems. Bounia- 

 kowsky, V. [1869] St. Pet. Ac. Sc. Bll. 14 

 (1870) 356-, 432-. 



of sums of products of first p - 1 numbers, 

 modulus p* or p 3 . Glaisher, J. W. L. QJ. 

 Mth. 31 (1900) 321-. 



Eoots of unity, properties. Ruffini, P. Mil. 



Mm. I. Lomb. Ven. 3 (1816-17) 67-. 

 Secondary roots of prime numbers, properties. 



Jonquieres, de. C. E. 122 (1896) 1513-. 

 Septimic residues. Pepin, T. C. E. 80 (1875) 



811-. 

 Wilson's and Fermat's theorems derived from 



common principle. Grunert, J. A. Crelle 



J. 8 (1832) 187-. 

 . Toeplitz, J. Grunert 



Arch. 32 (1859) 104-. 



, divisibility of faculties, etc. 

 Oettinger, L. Arch. Mth. Ps. 48 (1868) 

 159-. 



, proofs. Petersen, J. P. C. (xn) 



Ts. Mth. 2 (1872) 64-. 



theorem. Brennecke, (Dr) . Crelle J. 19 

 (1839) 319-. 



. Sylvester, J. J. Camb. and Dubl. Mth. 



J. 9 (1854) 84-. 

 . Liouville, J. Liouv. J. Mth. 1 (1856) 



351-. 

 . Catalan, E. C. Liege S. Sc. Mm. 15 



(1888) No. 1, 139-. 

 . Daublebsky von Sterneck, R. Mh. Mth. 



Ps. 7 (1896) 145-. 

 , extension. Bouniakowsky, V. [1856] 



St Pet. Ac. Sc. Bll. 15 (1857) 202-. 

 , to all numbers. Sylvester, J. J. (vi 



Adds.) Ph. Mg. 13 (1838) 454 ; 14 (1839) 47-. 



t Gauss's extension. Pollock, (Sir) F. 

 [1847] E. S. P. 5 (1851) 664-. 



, treatment. Terquem, O. N. A. Mth. 



2 (1843) 193-. 

 , . Arndt, F. Crelle J. 31 (1846) 



329-. 



, generalisation. Babbage, C. Edinb. 

 Ph. J. 1 (1819) 46-. 



, . Crelle, A. L. [1839] Berl. B. 



(1839) 133- ; Crelle J. 20 (1840) 29-. 

 , _. Scherk, H. F. D. Nf. Vsm. B. 



(1846) 204-. 

 , . Del Beccaro, T. Em. E. Ac. Line. 



Mm. 1 (1894) 344-. 



217 



