428 History of the Theory of Numbers. [Ch.u'. xviii 
-^ (2n+l)^n+2a — 1, where a is the difiference between 2/1+1 and the next 
greater prime. 
F. Stasi^^- noted that iV 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. Zondadari^^^ noted that 
Sin-TTX * TTX 
(7rx)^(l— x^)^n=2W sin irx/n 
is zero when j = ± p (pa prime) and not otherwise. 
A. Chiari'^"* cited known tests for primes, as the converse of Wilson's 
theorem. 
H. C. PockHngton^^^ employed single valued functions (}>(x), rp(x), 
vanishing for all positive integers a: (as = i/' = sin ttx), and real, finite and not 
zero for all other positive values of x. Then, for the gamma function F, 
^.(,)+^.(i±IM) 
is zero if and only if x is a prime [Wigert^^^"]. 
E. B. Escott^^^ stated that if we choose ai, . . . , a„, b so that the coeflBicients 
of x^", x^"~^, . . . , a:^ in the expansion of 
(x"+aix'*-^+...+aj2(x+5) 
are all zero, then all the remaining coefficients, other than the first and last, 
are divisible by 2?i + l if and only if 2/i + l is a prime. 
J. de Barinaga^^^ concluded from Wilson's theorem that if (P — 1)! is 
divided by 1+2+ . . . +(P-1) =P(P-l)/2, the remainder is P-1 when P 
is a prime, but is zero when P is composite (not excluding P = 4 as in the con- 
verse of Wilson's theorem). Hence on increasing by unity the least positive 
residues ?^0 obtained on di\'iding 1-2. . .x by 1+2+ . . . +x, for x= 1, 2, 
3, . . . , we obtain the successive odd primes 3, 5, ... . 
M. Vecchi^^° noted that, if x^ 1, A^>2 is a prime if and only if it be of 
the form 2^'— tt, where tt is the product of all odd primes ^p, p being 
the largest odd primed [\/iV], and where tt' is a product of powers of 
primes >p with exponents ^0. Again, A^> 121 is a prime if and only if of 
the form tt — 2V where y^l. 
Vecchi^^^ gave the simpler test: A^>5 is a prime if and only ii a—^=N, 
a+j8 = 7r, for a, /3 relatively prime, where tt is the product of all the odd 
primes ^[VA'']- 
G. Rados^^^ noted that p is a prime if and only if {2!3! . . . (p — 2)! 
(p-l)!j4=i(modp). 
CarmichaeP^ gave several tests analogous to those by Lucas. 
i«Il Boll. Matiraatica Giorn. Sc.-Didat., Bologna, 6, 1907. 120-1. 
"'Rend. Accad. Lined, (5), 19, 1910, 1, 319-324. i»^Il Pitagora, Palermo, 17, 1910-11, 31-33. 
"*Proc. Cambr. Phil. Soc., 16, 1911, 12. »»«L'interm6diaire des math., 19, 1912, 241-2. 
"'Revista de la Sociedad Mat. Espanola, 2, 1912, 17-21. 
"»Periodico di Mat., 29, 1913, 126-8. "»Math. 6s Term^s firtesito, 34, 1916, 62-70" 
i 
