Chap. XVIII] Tests FOR Primality. - 427 
L. Gegenbauer^^^ noted that 4n+l is a prime if 
L 4i/ J"L 4?/ J 
for every odd y, 1<?/^ V4n+1, and gave two similar tests for 4n+3. 
D. Gambioli^^^ and O. Meissner^^^ discussed the impracticabiUty of the 
test by the converse of Wilson's theorem. 
J. Hacks^^^ gave the characteristic relations for primes p: 
K. Zsigmondy^^^ noted that a number is a prime if and only if not 
expressible in the form aia2+i3i/32, where the a's and j3's are positive integers 
such that ai+a2=7/3i— 1S2. An odd number C is a prime if and only if 
C-\-k^ is not a square for k = 0, 1,. . ., [(C-9)/6]. 
R. D. von Sterneck^^^" gave several criteria for the (s + l)th prime by use 
of partitions into elements formed from the first s primes. 
H. Laurent^^^* noted that 
equals or 1 according as z is composite or prime. 
Fontebasso^^® noted that A^ is a prime if not divisible by one of the 
primes 2, 3, . . ., p, where iV/p<p+4. 
H. Laurent ^^^ proved that if we divide 
Fnix) = n\l -a:0(l -x^^) . . . (1 -x^""^'-'') 
3 = 1 
by (x" — l)/(a: — 1), the remainder is or n"~^ according as n is composite or 
prime. If we take a: to be an imaginary root of a;" = 1, Fn{x) becomes or 
n"~^ in the respective cases. 
Helge von Koch^^^ used infinite series to test whether or not a number is a 
power of a prime. 
Ph. Jolivald^^^ noted that, since every odd composite number is the 
difference of two triangular numbers, an odd number iV is a prime if and 
only if there is no odd square, with a root ^ (2iV — 9)/3, which increased by 
8A^ gives a square. 
S. Minetola^^" noted that, if k—n is divisible by 2n + l, then 2A;+1 is 
composite. We may terminate the examination when we reach a prime 
2n+l for which {k-n)/{2n+l)^n. 
A. Bindoni^^^ added that we may stop with a prime giving (k—n) 
isiSitzungsber. Ak. Wiss. Wien (Math.), 99, Ila, 1890, 389. 
is^Periodico di Mat., 13, 1898, 208-212. '"Math. Naturw. Blatter, 3, 1906, 100 
>8*Acta Mathematica, 17, 1893, 205. ""Monatsh. Math. Phys., 5, 1894, 123-8. 
"S'^Sitzungsber. Ak. Wiss. Wien (Math.), 105, Ila, 1896, 877-882. 
iss^Comptes Rendus Paris, 126, 1898, 809-810. "^guppj^ Periodico di Mat., 1899, 53. 
i"Nouv. Ann. Math., (3), 18, 1899, 234-241. 
"sOfversigt Veten.-Akad. Forhand., 57, 1900, 789-794 (French). 
"9L'interm4diaire des math., 9, 1902, 96; 10, 1903, 20. 
""11 Boll. Matematica Giorn. Sc.-Didat., Bologna, 6, 1907, 100-4. i"/6td., 165-6. 
