Chap. I] PERFECT, MULTIPLY Perfect, and Amicable Numbers. 21 
the impossibility of which is evident when the exponents j3, 7, . . . are other 
than 1, 0, 0, . . ., a case giving Euclid's solution [cf. Desboves^^']. 
C. G. Reuschle^^* gave in his table C the exponent to which 2 belongs 
modulo p, for each prime p<5000. Thus 2" — 1 has the factor 1399 for 
n = 233, the factor 2687 forn = 79, and 3391 for n = 113 [as stated exphcitly 
by Le Lasseur^^^'^^^]. ^iso 23514513 for n = 47, 1433 for w = 179, and 1913 
for n = 239. In the addition (p. 22) to Table A, he gave the prime factors of 
2^* — 1 for various n's to 156, 37 being the least n for which the decomposition 
is not given completely, while 41 is the least n for which no factor is known. 
For 34 errata in Table C, see Cunningham^^° of Ch. VII. 
F. Landry^^^ gave a new proof that 2^^ — 1 is a prime. 
Jean Plana^^° gave (p. 130) the factorization into two primes: 
2*^-1 = 13367X 164511353. 
His statement (p. 141) that 2^^ — 1 has no factor < 50033 was corrected by 
Landry^^^ (quoted by Lucas, "^ p. 280) and Gerardin."' 
Giov. Nocco^" showed that an odd perfect number has at least three 
distinct prime factors. For, if a"*6'* is perfect, 
2a- = V-T^' 6" = ^ -^, 
0—1 a— 1 
whence 
^ ^ Q^""^^ _ (a-l)b"+l 
2(5-1)" 2(6 -l)a"»" 6"+^-l ' 
a+fe(a6"+26"-'+2)=2+6(26"+2a6"-^). 
But the minunum values of a, h are 3, 5. Thus 6(a— 2)>2a — 2, 
a6''-26" = 6"-^-6(a-2)>6'*-^(2a-2), a6'»+26"-'>26"+2a6''-\ 
contrary to the earlier equation. In attempting to prove that every even 
perfect number 2'"6Vd' ... is of EucUd's type, he stated without proof that 
2-+16V. . . =(2"'+^-l)J5C. . ., B= \ / , C = - -,.. . 
— 1 c — 1 
require that 2"*+^ = B, 6" = 2^"+^ - 1 , d' = C, . . . (the first two of which results 
yield Euclid's formula). 
F. Landry^^^ stated (p. 8) that he possessed the complete decomposi- 
tion of 2"±l(n^64) except for 2^^±1, 2«Hl, and gave (pp. 10-11) the 
factors of 2^^-l and of 2"+l for n = 65, 66, 69, 75, 90, 105. 
"^Mathematische Abhandlung, enthaltend neue Zahlentheoretische Tabellen sammt einer 
dieselben betreflfenden Correspondenz mit dem verewigten C. G. J. Jacobi. Prog., Stutt- 
gart, 1856, 61 pp. Described by Kummer, Jour, fur Math., 53, 1857, 379. 
^°'Proc6des nouveaux pour demontrer que le nonabre 2147483647 est premier. Paris, 1859. 
Reprinted in SpLinx-Oedipe, Nancy, 1909, 6-9. 
""Mem. Reale Ac. Sc. Torino, (2), 20, 1863, dated Nov. 20, 1859. 
'"Alcune teorie su'numeri pari, impari, e perfetti, Lecce, 1863. 
"^Aux math^maticiens de toutes les parties du monde: communicatidn but la decomposition des 
nombres en leurs facteurs simples, Paris, 1867, 12 pp. 
