Odd Sublime Numbers?
Kevin Brown
of n, and let sigma(n) denote the sum of those divisors. The ancient
Greeks classified each natural number n as "deficient", "abundant", or
"perfect" according to whether sigma(n)-n was less than, greater than,
or equal to n.
A recent thread in sci.math has considered numbers n such that tau(n)
and sigma(n) are both perfect (viz, "n has a perfect number and sum
of divisors"). For brevity, let's call such numbers "sublime". The
only two sublime numbers I can find are 12 and
6086555670238378989670371734243169622657830773351885970528324860512791691264
Each of these examples is based on a Mersenne prime of the form
q = 2^k - 1 where k = 2^j - 1 is also a Mersenne prime and k-1 is
the sum of exactly j-1 distinct Mersenne exponents. The only known
primes q=2^k-1 with k=2^j-1 are given by k = 3, 7, 31, and 127.
However, (7-1) is not a sum of 2 Mersenne exponents, nor is (31-1) a
sum of 4 Mersenne exponents. Thus, the only two known sublime numbers
are based on the sums
(3-1) = 2
(127-1) = 61 + 31 + 19 + 7 + 5 + 3
Assuming there are no ODD perfect numbers (?), there can be no more
EVEN sublime numbers unless there are other (presently unknown) Mersenne
prime exponents that are themselves Mersenne primes. Given that
the exponents 131071 and 524287 have already be checked, the next
possible exponent would be 2^31 - 1, which is 2,147,483,647. Even
with the Lucas-Lehmer test I think this exponent would be a challenge.
Notice that we would not only need to test this particular exponent,
but all exponents less than this one, in order to decide if this number
minus 1 was expressible as a sum of distinct Mersenne exponents.
The next possible exponent would be 2^61-1, which is certainly
far out of reach for any known method of testing.
Anyway, it remains to consider the possibility of an ODD sublime
number (again assuming no odd perfect numbers). By the same arguments
presented in a previous post addressing EVEN sublime numbers, we know
that an odd sublime number must be of the form
N = (q^r)(p1 p2 ... pt)
where r = 2^j - 2 is one less than a Mersenne prime, t = j - 1, and
the pi = 2^ki - 1 are distinct odd Mersenne primes and q is an odd
prime. Also, we must have
q^(r+1) - 1
----------- = 2^k0 - 1 (prime)
q - 1
where (k0 - 1) = k1 +k2 +... + kt. To give a "non-example", consider
the case of q=5,j=2. This gives r=2, so the above expression is
(5^3-1)/(5-1) = 31, which is a prime of the form 2^k0 - 1 with k0=5.
Therefore, we need only express (5-1)=4 as a sum of (j-1) distinct
Mersenne exponents. Since j-1 equals 1 in this case, it requires
that 4 itself be a Mersenne exponent, i.e., we need 2^4-1 = 15 to
be a prime, which of course it isn't. If it was, the number
N = (5^2)(15) = 375 would have tau(N)=6 and sigma(N)=496, and
so 375 would be an odd sublime number.
The above shows that the first step to find an odd sublime number
is to find an odd prime q and two Mersenne primes m1=2^j-1 and m2=2^k-1
such that (q^m1 - 1) = (q-1) m2. If we can find such primes, then we
need to express k-1 as a sum of exactly j-1 distinct Mersenne exponents.
There may be a simple way of proving this is impossible, but off hand
I don't see it.
By the way, for the amusement of people interested in numerological
coincidences, the two (known) sublime numbers offer plenty of
opportunity. The number 12 (as in the twelve tone musical scale,
the twelve months of the year, the twelve apostles, the twelve hours
on a clock, the twelve signs of the zodiac, etc) needs no introduction.
People familiar with the "large number coincidence" in physics may
be entertained by the value of the 2nd (known) sublime number
s2 = (6.086)10^75 sqrt(s2) = (7.8)10^37
Also, if s2 is factored into odd and even parts s2 = (o2)(e2) we find
/ e2 \ 4
( ---- ) = (2) (0.99935...)
\ o2 /