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] ,,-i.-iLpJ -, 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 y-i 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 ^ (2JV—9)/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 (k—ri) "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. 1M«Sitzungsber. 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.