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Olivier Gérard ogerard at
Wed Apr 18 16:40:43 CEST 2001

Hello Simon,

On Wed, Apr 18, 2001 at 03:14:11PM +0100, Simon Colton wrote:
> Dear SeqFans,
> I thought you might be interested in another cute theorem produced
> by my HR program recently.

> The binary representation of every perfect number has a prime number
> of 1s followed by zeros.
> This result can be proved fairly painlessly using Hardy and Wright's
> theorem 277. Actually, a stronger version is:
> ------------------------------------------------------------------------------
> The binary representation of the n-th perfect number consists of the
> n-th Mersenne prime (call this k) lots of ones, followed by k-1 zeros.
> ------------------------------------------------------------------------------
> I like this result because (a) a friend of mine (Jeremy Gow) submitted
> pernicious numbers to the Encyclopedia (b) the fact that perfects are pernicious
> is appealing and (c) I would never have thought about looking at the binary
> representation of perfect numbers.

I like this result too, and not only because I created the nialpdromes,
as cousins of my former katadromes. It is another example of the interest of the EIS, but

c) looking at the binary representation of a perfect number as well as the final
result, could have been found by people knowing (and they are many) that every EVEN perfect
number is constructed from a Mersenne prime and that a mersenne prime derives from
a perfect power of 2, so may have a remarquable binary representation.
(I think that the demonstration of the link between even perfect numbers and Mersenne primes
is due to Euler)

What would be more interesting would be whether this emphasis on binary representation
could furnish an argument against the existence of an ODD perfect number.
I doubt it, though. 

I don't know of recent progress for this conjecture, one of the latest result being
that an odd perfect number is at least above 10^300 and that the largest prime is > 500000
and with a total power sum (prime factors with multiplicity) of at least 29 .

R.P. Brent, G.L. Cohen, H.J.J. te Riele, Improved Techniques for lower bounds for odd
perfect numbers, Mathematics of Computation, 57 (1991), p857-868.
I think there is also an article by M. Brandstein.

Another problem is : is there an infinity of Mersenne primes ?
It seems so, but we don't know.

> I imagine that this, or a more general, result is well known, and would
> very much appreciate any references.

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