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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

633.0. "Perfect numbers" by VERDI::LEMIEUX () Fri Dec 19 1986 20:27

    I need to know the 5th perfect number, the one that comes after
    8128.
    
    kevin
T.RTitleUserPersonal
Name
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633.1Easy enough to check it out...OMEGA::REILLYFri Dec 19 1986 21:176
    I don't know why, but I seem to remember that it is 33,550,336
    but then again, I may be wrong.....                           
    
    						matt
    (then again, it may have been the combination to a gym locker, 
    the number sticks in my head for some reason....)
633.2More rigorous methodOMEGA::REILLYFri Dec 19 1986 21:5534
    
For what it is worth....

       	       8128 = 2 ^ 6 * (2 ^ 7 - 1)
       
     Since 2 ^ 7 - 1 is prime, 8128 is perfect.  (This is easily proved 
by someone who is suitably awake.)

     All of the known perfect numbers are of the form 
2 ^ (p - 1) * (2 ^ p - 1)  where p is prime and (2 ^ p - 1)
is prime.
       
isprime(2^11 - 1);
                                     false

isprime(2^13 - 1);
                                      true

evalf(2^12 * (2 ^ 13 - 1));
                                   33550336.

quit;

       	       	       	       	       	       	    matt

btw, for grins try the above commands to MAPLE in a batch job 
without the quit command at the end.... i.e. the following...

$ maple -q
isprime(2^11 - 1);
isprime(2^13 - 1);
evalf(2^12 * (2 ^ 13 - 1));
convert(2^12 * (2 ^ 13 - 1),  binary);
    
633.3More perfect numbersMODEL::YARBROUGHMon Dec 22 1986 11:2623
The next few perfect numbers are 2^(k-1) * (2^k-1) for 
k=17	8589869056
k=19	137438691328
k=31	2305843008139952128
k=61
k=89
k=107
k=127
k=521
k=607
k=1279
k=2203
k=2281
k=3217
k=4253
k=4423
k=9689
k=9941	(5985 digits)

The primes of the form 2^k-1 are the corresponding Mersenne primes.

The information above is taken from Beiler's *Recreations in the Theory of 
Numbers*, p19.