000RM.dvi

24.2 Charles Twigg on the first 10 perfect numbers 629

24.2 Charles Twigg on the first 10 perfect numbers

There are only 39 known Mersenne primes, and therefore 39 known per-
fect numbers. See Appendix. LetPnbe then-th perfect number.

nk M 123 k 6 Pn=2k−^1 Mk

(^237) 3 5 31 (^28496)
4 7 1275 13 8191 (^812833550336)
6 17 1310717 19 524287 (^8589869056137438691328)
8 31 21474836479 61 2305843009213693951 (^23058430081399521282658455991569831744654692615953842176)
10 89 618970019642690137449562111 191561942608236107294793378084303638130997321548169216

• P 1 is the difference of the digits ofP 2 .InP 2 , the units digit is the
  cube of the of tens digit.


• P 3 andP 4 are the first two perfect numbers prefaced by squares.
  The first two digits ofP 3 are consecutive squares. The first and last
  digits ofP 4 are like cubes. The sums of the digits ofP 3 andP 4 are
  the same, namely, the prime 19.


• P 4 terminates bothP 11 andP 14.^2


• Three repdigits are imbedded inP 5.


• P 7 contains each of the ten decimal digits except 0 and 5.


• P 9 is the smallest perfect number to contain each of the nine nonzero
  digits at least once. It is zerofree.


• P 10 is the smallest perfect number to contain each of the ten decimal
  digits at least once.

(^2) These contain respectively 65 and 366 digits.