Chap. IJ PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 25 
communication from Pellet, 2" — 1 is divisible by 6n+l if n and 6n+l are 
primes such that 6n+l =4L^+27M^ [provided* n= 1 (mod 4), i. e., L is odd]. 
M. A. Stern^" amplified Euler's^^ proof concerning odd perfect numbers. 
E. Lucas^^^ repeated the statement [Desboves^^^] that an odd perfect 
number must contain at least four distinct primes. 
G. Valentin^^^ gave a table, computed in 1872, showing factors of 
2"— 1 for n = 79, 113, 233, etc., but not the new cases of LeLasseur.^^^ 
The primality of iV = 2®^ — 1, a number of 19 digits, considered composite 
by Mersenne and prime by Landry, was established by J. Pervusin"° and 
P. Seelhoff ^^^ independently. The latter claimed to verify that there is no 
factor <N^'^ of the form 8?i+7, abbreviating the work by use of various 
numbers of which iV is a quadratic residue; thus iV is a prime or the product 
of two primes. Since iV = 2(2^°)^ — 1, 2 is a quadratic residue of any prime 
factor of N, so that the factor is 8n=i= 1. It was verified that 3^= 1 (mod N), 
where i8 = (iV — 1)/9. If N=fF, where F is the prime factor 8n-|-l, then 
3^^1(mod F) and, by Fermat's theorem, 3^~^=l(mod F). It is stated 
without proof that one of the exponents /S and F — 1 divides the other. 
Cole^^^ regarded the proof as unsatisfactory. 
Seelhoff proved that a perfect number of the form pV is of Euclid's 
type if p and r are primes and p<r. The condition is 
r''+\2-p)-2r''{l-p)-p 
If p > 2, the denominator is negative. Hence p = 2 and 
^'=2K^' 2'+' = r+;j-j, p = l, r=2-+'-l. 
His statements (p. 177) about the factors of 2" — 1, n = 37, 47, 53, 59, 
were corrected by him {ibid., p. 320) to accord with Landry.^^^ 
P. Seelhoff ^^^ obtained the known factors of these 2" — 1 and proved that 
2^^ — 1 is a prime, by use of his method of quadratic residues. 
H. Novarese^^^ proved that every perfect number of Euclid's type ends 
in 6 or 28, and that each one > 6 is of the form 9A;+1. 
Jules Hudelot"^ verified in 54 hours that 2^^ — 1 is a prime by use of the 
test by Lucas, Recreations math., 2, 1883, 233. 
♦Correction by Kraitchik, Sphinx-Oedipe, 6, 1911, 73; Pellet, 7, 1912, 15. 
"'Mathesis, 6, 1886, p. 248. 
"8/6td., p. 250. 
""Archiv Math. Phys., (2), 4, 1886, 100-3. 
""Bull. Acad. Sc. St. Petersb., (3), 31, 1887, p. 532; Melanges math. astr. ac. St. P^tersb., 6, 
1881-8, 553; communicated Nov. 1883. 
"iZeitschr. Math. Phys., 31, 1886, 174-8. 
»"Archiv Math. Phys., (2), 2, 1885, 327; 5, 1887, 221-3 (misprint forn = 41). 
*«Jomal de sciencias math, e astr., 8, 1887, 11-14. [Servais"*.] 
"♦Mathesis, 7, 1887, 46. Sphinx-Oedipe, 1909, 16. 
