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PERFECT-NUMBERS.Pdf PERFECT NUMBERS A Perfect Number N is defined as any positive integer where the sum of its divisors minus the number itself equals the number. The first few of these, already known to the ancient Greeks, are 6, 28, 496, and 8128. Euclid, over two thousand years ago, showed that all even perfect numbers can be represented by – N=2 p-1(2 p-1) where p is a prime for which 2 p-1 is a prime number That is, we have an even Perfect Number of the form N whenever the Mersenne Number 2p-1 is a prime. Undoubtedly Mersenne was familiar with Euclid’s book in coming up with his primes. The following gives a table of the first nine Mersenne Primes and Perfect Numbers- Prime, p Mersenne Prime, 2 p-1 Perfect Number, 2 p-1(2 p-1) 2 3 6 3 7 28 5 31 496 7 127 8128 13 8191 33550336 17 131071 8589869056 19 524287 137438691328 31 2147483647 2305843008139952128 61 2305843009213693951 2658455991569831744654692615953842176 What is quite clear is that, once a Mersenne Prime has been found, the corresponding Perfect Number is known and visa-versa. Furthermore, as the table shows, the Mersenne Primes always end in 1 or 7 and the Perfect Numbers in 6 or 8. This observation at once calls one’s attention to a mod operation. Indeed, we find that Mersenne Primes M generated by primes p>2 and Perfect Numbers N generated by primes p>3 have the following properties- ( 2 p-1) mod(8)=7 and [2p-1(2 p-1)] mod(8)=0 These results actually apply to all positive integers not just primes and have their origin in the exponential forms of the two expressions. Thus (2 3456 -1) mod(8)=7 and (2 320 (2 321 -1))mod(8)=0. Recalling our earlier results on the integer spiral for all positive integers, we can state that all Perfect Numbers above 28 fall along the positive x axis as shown- In the graph we have also plotted the lines along which Mersenne Primes (2 p-1) and Fermat Primes (2 2^p+1) are located. The figure also suggests that other primes of the form 2p+3 and 2 p+5 are to be found along the remaining two diagonals and also some pseudo-perfect numbers along the y axis and the negative x axis. The number of Perfect Numbers as well as the Mersenne Primes appear to be infinite although this needs to still be proven. The facts that there are no Fermat primes above 65537 and that no odd Perfect Numbers exist, need also to be proven. We note that the perfect numbers 6=1+2+3 and 28=1+2+4+7+14 have number fractions f[6]=(2+3)/6=5/6 and f[28]=(2+4+7+14)/28=27/28. This suggests that we can always generate Perfect Numbers by use of the number fraction concept discussed in an earlier note. The starting point for doing so is the equality- ∑divisors of N − (N + )1 (N − )1 f [N ] = ≡ N N This is equivalent to saying- b = ∑divisors of N − 2N = 0 which also agrees with the basic definition for a Perfect Number. In computer language, this result is expressed as- with(numtheory): for N from 1 to 100 do{N,(( add(i,i=divisors(N))-(2*N)))}od; and produces an N when b=0. Just to demonstrate how this type of evaluation works, consider the case of M=2047=2 11 -1=23 x 89. This number is a Mersenne Number but not a Mersenne Prime. Without knowing this beforehand, an evaluation of b yields b= -1934 and hence M cannot be a prime. Another advantage of the b evaluation is that it can be used to work things backwards starting with a Perfect Number N and deducing the value of its corresponding Mersenne Prime M. The procedure works as follows- (1)-Run the above program for b over a range of Ns until a value b=0 is found (2)-Use the value of N for which this is the case and run N/2 (p-1) over a range of low value primes p until the quotient assumes its lowest odd integer value (3)-This produces a Mersenne Prime M corresponding and the corresponding prime p. To demonstrate this approach, consider running the computer program over the range 3355030 <N<3355040 . This produces a value of b=0 at N=3355036 and a division of N/2 (p-1) yields an odd integer value M= 8191 at p=13. That is- M=213 -1=8191 is a Mersenne Prime. U.H.Kurzweg Feb. 2013 .
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