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TESTS FOR FRUGALITY.                               427
L. Gegenbauer181 noted that 4w+l is a prime if ptn+l-i/n r4n-3-~3/21 L 4y J"L 40 J
for every odd y, Ky^ V4n+l, and gave two similar tests for 4n+3.
D. Gambioli182 and 0. Meissner183 discussed the impracticability of the test by the converse of Wilson's theorem.
J. Hacks184 gave the characteristic relations for primes p:
'* -51 par]
2 / v//    ''      ^\ ,.! LpJ   -iLyJJ    \ 2 K. Zsigmondy186 noted that a number is a prime if and only if not
expressible in the form a1a2+j8ij32, where the a's and /3's are positive integers
such that ai+cfc =&--&.   An odd number C is a prime if and only if
C+fc2 is not a square for & = 0, 1,. . ., [(C-9)/6].
R. D. von Sterneck186" gave several criteria for the (s+l)th prune by use
of partitions into elements formed from the first s primes. H. Laurent1866 noted that
equals 0 or 1 according as z is composite or prime.
Fontebasso186 noted that N is a prime if not divisible by one of the primes 2, 3,. . ., p, where N/p<p+4t.
H. Laurent187 proved that if we divide
by (xn  !)/( 1), the remainder is 0 or n"""1 according as n is composite or prime. If we take x to be an imaginary root of x" = 1, Fn(x) becomes 0 or n"^1 in the respective cases.
Helge von Koch188 used infinite series to test whether or not a number is a power of a prime.
Ph. Jolivald189 noted that, since every odd composite number is the difference of two triangular numbers, an odd number N is a prime if and only if there is no odd square, with a root ^ (2JV9)/3, which increased by &Y gives a square.
S. Minetola190 noted that, if k n is divisible by 2n+l, then 2&+1 is composite. We may terminate the examination when we reach a prime 2n+l for which (t-n)/(2n+l)5n.
A. Bindoni191 added that we may stop with a prime giving (kri)
"iSitzungsber. Ak. Wiss. Wien (Math.), 99, Ila, 1890, 389.
1MPeriodico di Mat., 13, 1898, 208-212.                     1MMath. Naturw. Blatter, 3, 1906, 100
1MActa Mathematica, 17, 1893, 205.                        1MMonatsh. Math. Phys., 5, 1894, 123-8.
1MSitzungsber. Ak. Wiss. Wien (Math.), 105, Ila, 1896, 877-882.
1MbComptes Rendus Paris, 126, 1898, 809-810.          1MSuppl. Periodico di Mat., 1899, 53.
"7Nouv. Ann. Math., (3), 18, 1899, 234r-241.
5fveraigt Veten.-Akad. Fftrhand., 57, 1900, 789-794 (French).
"L'intermfidiaire des math., 9, 1902, 96; 10, 1903, 20.
I1 Boll. Matematica Giorn. Sc.-Didat., Bologna, 6, 1907, 100-4.          1M/6u*., 165-6.