428 HlSTOKY OF THE THEORY OF NUMBERS. [CHAP. XVHI -K2n+l)gn+2a— 1, where a is the difference between 2n+l and the next greater prime. F. Stasi192 noted that AT is a prime if not divisible by one of the primes 2, 3, . . ., p, where N/p<p+2a and a is the difference between p and the prime just >p. E. Zondadari193 noted that sin irx/n is zero when x = =*=p (pa prime) and not otherwise. A. Chiari194 cited known tests for primes, as the converse of Wilson's theorem. EL C. Pocklington195 employed single valued functions 4>(z), iKs), vanishing for all positive integers x (as <t> = \l/ = sin TTX) , and real, finite and not zero for all other positive values of x. Then, for the gamma function F, is zero if and only if x is a prime [Wigert2360]. E. B. Escott196 stated that if we choose <h, . . . , any b so that the coefficients of x2n, x2n~2, . . . , x2 in the expansion of are all zero, then all the remaining coefficients, other than the first and last, are divisible by 2n-\-l if and only if 2n+l is a prime. J. de Barinaga197 concluded from Wilson's theorem that if (P— l)!is divided by 1+2+ . . . + (P-1) = P(P-l)/2, the remainder is P-l when ? is a prime, but is zero when P is composite (not excluding P =4 as in the converse of Wilson's theorem) . Hence on increasing by unity the least positive residues ^0 obtained on dividing 1-2. . .x by 1+2-J-. . . +x, for x= 1, 2, 3, . . . , we obtain the successive odd primes 3, 5, .... M. Vecchi140 noted that, if x*z 1, N>2 is a prime if and only if it be of the form 2V — TT, where w is the product of all odd primes ^p, p being the largest odd prime ^[VF], and where ir' is a product of powers of primes > p with exponents ^ 0. Again, N> 121 is a prime if and only if of the form IT— 2V where y^l. Vecchi198 gave the simpler test: 7V>5 is a prime if and only if a— /3=JV, a-j-/5=7rj for a, j8 relatively prime, where w is the product of all the odd prmes G. Rados199 noted that p is & prime if and only if {213! ... (p -2)1 Carmichael93 gave several tests analogous to those by Lucas. 1MI1 Boll. Matimatica Giorn. Sc.-Didat., Bologna, .6, 1907. 120-1. 1MRend. Accad. Lincei, (5), 19, 1910, 1, 319-324. 1MI1 Pitagora, Palermo, 17, 1910-11, 31-33. 1MProc. Cambr. Phil. Soc., 16, 1911, 12. lML'interm6diaire des math., 19, 1912, -241-2. 197Revista de la Sociedad Mat. Espafiola, 2, 1912, 17-21. 198Periodico di Mat., 29, 1913, 126-8. 188Math. 4s Termed ErtesitO, 34, 1916, 62-70'