CHAP, ij PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 37
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E. Lucas320 gave a table of Pm of the form 2n~1(2n-l)AT which includes only 15 of the 26 Pm given above .and no additional Pm> m>2, except two erroneous P5:
210345-72412.19.23-89, 2175.7213-19237.73-127,
attributed elsewhere321 by him to Fermat. If we replace 72 by 7 in the former, we obtain a correct P5 listed by Carmichael :332
P5(7>=210345.7.11219.23-89.
If in the second, we replace 5-72 by 35-5-73 we obtain Fermat;s P5(6).
A. Desboves322 noted that 120 and 672 are the only P3 of the form 2n-3-p, where p is a prime.
D. N. Lehmer323 gave the additional Pm:
P4(12)=22325-7213-19,
P4(13) = 28327213-19237-73-127,
P5(8; = 22136527.19.23231.79-89-137.547.683.1093,
pQ(4) = 21936537211.13-19.23-3141.137.547.1093,
P6(5) = 224385.72114347.1923143.53-127.379-601-757.1801.
He readily proved that a P3 contains at least 3 distinct prime factors, a P4 at least 4, a P5 at least 6, a P6 at least 9, a P? at least 14.
J. Westlund324 proved that 233-5 and 253-7 are the only P3 of the form PiaP2bPz, where the p's are primes and pi<p2<Ps- He325 proved that the only PB^P^PBP*, P1<P2<P3<P*, is P3(3) = 293.11.31.
A. Cunningham326 considered Pm of the form 2q~~l(29-l)F, where F is to be suitably determined. There exists at least one such Pm for every q up to 39, except 33, 35, 36, and one for q = 45, 51, 62. Of the 85 Pm found, the only one published is the largest one, viz., for g = 62, giving P6(6) with
F = 37547211.13.19223-59.71-79427.157.379-75743331.3033169;
while none have m>6, and for ?n = 3 at most one has a given q. He found in 1902 (but did not publish) the two P7 = 246(247-1)F, where
F = C-192127or C-194151-911,
C = 315-53-75-ll-13-17-23-31-3741-43-61-89-97-193-442151.
R. D. Carmichael328 has shown that there exists no odd Pm with only three distinct prime factors; that 233-5 and 253-7 are the only Pm with only
320Bull. Bibl. e Storia Mat. e Fis., 10, 1877, 286. In 253-5-7, listed as a P4t 3 is a misprint for 3». '"Lucas, Th6orie des Nombres, 1, Paris, 1891, 380. Here the factor IP 133 of Fermat'a P«<1>
is given erroneously as 1M32, while the P&w of Descartes is attributed to Fermat. 322Questions d'Algebre, 2d ed., 1878, p. 490, Ex. 24. 32'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.