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CHAP. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 49
Ge*rardin382 proved that the three numbers (1) with n=m+2 are not all primes if 34< m g 60, the cases m = 38 and 53 not being decided. Replacing m by w+1 and k by 20+1 in case (1J of Euler364, we get the pair 2npq, 2nr, where n=m+2g+2,
with P=22H~1+1. For ^=0, we have the case (1) just mentioned; all values rag 200 are excluded except m=38, 74, 98, 146, 149, 182, 185, 197. The case g= 1 is excluded since y or z is a difference of two squares. For g = 2, all values w ^ 60 are excluded except m='29, 34, 37, 49. For g = 3, all values <100 are excluded except m=8, 15, 23, 92.
0. Meissner,383 using the notation of Cunningham,379 noted that n and s(n) are amicable if s2(n)=n and raised the question of the existence of numbers n for which sk(ri)=n for fc^3, so that n, s(ri),. . .^"^(n) would give amicable numbers of higher order. He asked if the repetition of the operation s, a finite number (k) of times always leads to a, prime, a perfect or amicable number; also if k increases with n to infinity. On these questions, see Dickson386 and Poulet.387
A. Ge*rardin384 stated that the only values n<200 for which the numbers (1) are all primes are the three known to Descartes.
L. E. Dickson385 obtained the two new pairs of amicable numbers
2M2959-50231, 24- 17- 137-262079; 24- 10103-735263, 24- 17- 137-2990783,
by treating the type 16pq, 16-17-137r, where p, g, r are distinct odd primes. These are amicable if and only if
p=m+9935, g=n+9935, r = 4(m+n) +88799, mn = 27347-23-73.
Although Euler364 mentioned this type (33) in §95, he made no discussion of it since r always exceeds the limit 100000 of the table of prunes accessible to him. An examination of the 120 distinct cases led only to the above two amicable pairs.
Dickson386 proved that there exist only five pairs of amicable numbers in which the smaller number is <6233, viz., (1), (a), (0), (60) in Euler's364 table, and Paganini's372 pair. In the notation of Cunningham,379 the chain n, s(ri), s2(n)j ... is said to be of period k if sk(ri) =n. The empirical theorem of Catalan374 is stated in the corrected form that every non-periodic chain contains a prime and verified for a wide range of values of n. In particular, if rt<6233, there is no chain of period 3, 4, 5, or 6. For k odd and >1, there is no chain an1} cm2, . . ., ank of period k in which n1}. . ., nk have no common factor and each n^ is prime to a> 1.
382Sphinx-Oedipe, 1907-8, 49-56, 65-71; some details are inaccurate, but the results correct. »MArchiv Math. Phya., (3), 12, 1907, 199; Math.-Naturw. Blatter, 4, 1907, 86 (for fc»3). 8MAssoc. frang. avanc. sc., 37, 1908, 36-48; I'interm^diaire des math., 1909, 104. 3MAmer. Math. Monthly, 18, 1911, 109. ""Quart. Jour. Math., 44, 1913, 264-296.