In Unsolved Problems in Number Theory, Richard K Guy says of Mersenne
primes: "their number is undoubtedly infinite, but proof is hopelessly
beyond reach".
He then offers some suggestions for the size of M(x), the number of
primes p <= x for which 2^p -1 is prime.
Gillies suggested M(x) ~ c ln x
Pomerance suggested M(x) ~ c ( ln ln x ) ^ 2
This is very serious indeed, especially for those of us who believe the
number of Mersenne primes to be finite.
It's a fairly old book in a manner of speaking: in 1981 he poses the
question as to whether 2^p - 1 is always square-free. I'm sure this has
been discussed here from time to time - did we ever get an answer? In
this case, Guy believes that the answer is no, and that it could be
settled by computer if you were lucky.
I'm still on topic if I talk about perfect numbers, where the sum of the
factors of a perfect number n, which I call s(n) is equal to n.
However I'm off topic as soon as I start talking about amicable numbers,
sometimes called semi-perfect numbers. For a pair of amicable numbers m
and n we have s(m) = n and s(n) = m. For example s(220) = 284 while
s(284) = 220.
I choose to mention these because of the recent mention of hairy and
smooth numbers and in the context of Esau and Jacob, also recent players
here, as the number 220 is of some significance in their story in
Genesis 32:14.
The recent heroes in this field are H J J te Riele, who "knows
everything about amicable numbers" according to a now forgotten usenet
poster and Lee and Madachy, who published "The history and discovery of
Amicable Numbers" in the Journal of Recreational Mathematics in 1972,
along with an alarmingly long list of then known amicable numbers. (Does
anybody know if this journal is still published? When I subscribed to it
for a while, though, it wasn't too recreational, and seemed obsessed
with repunits for a while.)
There are far more amicable pairs known than even perfect numbers, yet
Guy's claim on their infinite number or otherwise is, surprisingly,
weaker. "It is not known if there are infinitely many, but it is
believed that there are."
Finally, pi. Along with others, I have been amused by the reputed
Alabama legislature decision, and spend a lot of time looking at the
urban legends at http://www.snopes.com/ which is one of the most
significant sites on the web, possibly second only to
http://www.mersenne.org/prime.htm ?
However, as I believe in the inerrancy of scripture, I obviously have a
problem with 1 Kings 7:23. I don't believe either that pi = 3 or that
God thinks pi = 3. So, what happens? At
http://www.khouse.org/articles/biblestudy/19980401-158.html we can learn
that there is a subtle difference in the text from what might be
expected
in that the word for circumference "qav" has been replaced by the word
"qaveh". If we take note of the numerical values associated with these
words, qav = 100 + 6, while qaveh = 100 + 6 + 5. Accordingly, we take
the implied multiplicand of 3 and extend it by 111/106, which gives an
approximation of 333/106, which is 3.141509... which is accurate enough
for practical purposes. Possibly not for rocket science, but that's not
what we're talking about here. K House probably don't phrase their
explanation in the way I would choose, but it nevertheless makes
compelling reading from a reasonably mainstream source.
Over history, there have been numerous other approximations to the value
of pi. Our current culture seems to favour 22/7 as an approximation, and
the Biblical approximation above suggests 333/106. However, this is not
the best available in three digits, which is, so far as I know, 355/113,
which is correct to an astonishing one part in ten million. I understand
that in certain quarters, 3 1/7 was not in vogue, with 3 1/8 favoured.
What, argued these particular mystics, could be a better number than
five squared shared by two cubed? N P Smith asked whether we should be
more concerned by those who serious propose answers which are clearly
wrong or by those who spend time in repeatedly refuting these spurious
claims.
As for squaring the circle, another popular pastime, the Greeks noted
that a square of side 8 have pretty much the same area. This points to
256/81 or sixteen squared shared by nine squared if you like that sort
of thing. It's still not exact. That's what irrational means...
I'm sorry to have strayed off topic: at the moment I can't find any
legitimate connection between pi and Mersenne numbers - if anybody can
do so then obviously I am absolved because this posting will have been
on topic after all.
I am absolved! Between researching this article and posting it, others
have started to explore the possibility of such links.
Regards,
Ian W Halliday
Wellington, New Zealand
---
Happiness is just around the corner. - D H Lehmer
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