Chap. I] PERFECT, MULTIPLY PERFECT, AND AmICABLE NUMBERS. 37 
E. Lucas^^o ga^g a table of P^ of the form 2''-\2''-l)N which includes 
only 15 of the 26 Pm given above and no additional P^, m>2, except 
two erroneous P5 : 
2^°3^5-72.1l2.19.23.89, 2i^5-72l3.19237-73-127, 
attributed elsewhere^^^ by him to Fermat. If we replace 7^ by 7 in the 
former, we obtain a correct P5 listed by Carmichael 'P"^ 
P5(7> = 2'°3*5-7. 11219.23-89. 
If in the second, we replace 5-7^ by 3^-5-7^ we obtain Fermat's P^-^K 
A. Desboves^^^ noted that 120 and 672 are the only P3 of the form 
2"-3-p, where p is a prime. 
D. N. Lehmer^^^ gave the additional P„: 
p^(i2) ^22325.7213.19, 
P^(i3) = 2«3272l3.19237.73-127, 
P5® =2213^527.19.23231.79.89.137.547.683.1093, 
Pg(4) =2193^5^7211. 13.19.23.31.41. 137.547.1093, 
Pg(5) =22^3^5.7211.13.17.19231.43.53.127.379.601.757.1801. 
He readily proved that a P3 contains at least 3 distinct prime factors, a 
P4 at least 4, a P5 at least 6, a Pq at least 9, a P^ at least 14. 
J. Westlund324 proved that 2^3.5 and 2^3.7 are the only P3 of the form 
V\'V2Vzy where the p's are primes and Pi<P2<P3. He^^^ proved that the 
only P3 = Pi"p2P3P4, Pi<P2<P3<P„ is P3^'^ =293.11.31. 
A. Cunningham^^^ considered P^ of the form 2^ \2^—l)F, where F is 
to be suitably determined. There exists at least one such P^ for every q 
up to 39, except 33, 35, 36, and one for g = 45, 51, 62. Of the 85 P^ found, 
the only one published is the largest one, viz., for q = Q2, giving Pq^^^ with 
F = 3^5'72ll. 13.19223.59.71.79.127.157.379.757.43331.3033169; 
while none have m>6, and for m = 3 at most one has a given q. He found 
in 1902 (but did not publish) the two P7 = 2^H2^^-1)P, where 
F = C.192127 or 0.19^51-911, 
C = 3i^-5^.7^.1M3.17-23.31-37-41.43.61.89.97-193.442151. 
R. D. CarmichaeP^^ has shown that there exists no odd P^ with only 
three distinct prime factors; that 2^3-5 and 2^3.7 are the only P^ with only 
'20Bull. Bibl. e Storia Mat. e Fis., 10, 1877, 286. In 253-5-7, listed as a Pt, 3 is a misprint for 3». 
'"Lucas, Theorie des Nombres, 1, Paris, 1891, 380. Here the factor 11^ IS^ of Fermat's P«(') 
is given erroneously as lllS^, while the Pe^i^ of Descartes is attributed to Fermat. 
'^Questions d'Algebre, 2d ed., 1878, p. 490, Ex. 24. 
'^'Annals of Math., (2), 2, 1900-1, 103-4. 
"^Annals of Math., (2), 2, 1900-1, 172-4. 
'"Annals of Math., (2), 3, 1901-2, 161-3. 
'"British Association Reports, 1902, 528-9. 
'"American Math. Monthly, 13, Feb., 1906, 35-36. 
717 8 5 
