From Newsgroup: rec.arts.sf.fandom


15 September 2025

While counting my calf raises tonight, it struck me that two
to the third was followed immediately by three to the
second.

I wonder what pairs like this are called?

I've long been aware that sixteen is its own whatsit: it's
both four to the second and two to the fourth. Sixteen must
be the only such number, aside from numbers are a number
raised to itself.

I'll bet that those numbers also have a name. The series
1^1, 2^2, . . . has a rising rate that puts factorials to
shame.
--
Joy Beeson
joy beeson at centurylink dot net
http://wlweather.net/PAGEJOY/

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From Newsgroup: rec.arts.sf.fandom

Joy Beeson <jbeeson@invalid.net.invalid> wrote:

While counting my calf raises tonight, it struck me that two to the
third was followed immediately by three to the second.

I wonder what pairs like this are called?

Perfect powers. Catalan's Conjecture -- which has since been
proven -- is that 8 and 9 are the only adjacent ones. See https://oeis.org/A001597

Note that 2025 is a perfect power, the first since 1936 and the last
until 2048.

I've long been aware that sixteen is its own whatsit: it's both
four to the second and two to the fourth. Sixteen must be the only
such number, aside from numbers are a number raised to itself.

Any perfect power whose exponent isn't prime is also the perfect power
with every of exponent of that exponent's divisors. For instance 64
is not just 2^6 but also 4^3 and 8^2, since 6=3*2.

Once when someone asked me how old I was, I told them that my age in
days that week was both a square and a cube. That meant that my age
was also a sixth power, and there was only one plausible age in days
that was a sixth power for me, since I'm obviously neither a child nor
the oldest person who ever lived.

I'll bet that those numbers also have a name. The series
1^1, 2^2, . . . has a rising rate that puts factorials to
shame.

Apparently not, other than n^n, but you can read all about it at https://oeis.org/A000312

oeis.org is the Online Encyclopedia of Integer Sequences, founded 61
years ago, and still run by its founder. You can search within it for
my name to read about my humble contributions to it.
--
Keith F. Lynch - http://keithlynch.net/
Please see http://keithlynch.net/email.html before emailing me.
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From Newsgroup: rec.arts.sf.fandom

Keith F. Lynch <kfl@KeithLynch.net> wrote:

Joy Beeson <jbeeson@invalid.net.invalid> wrote:

I'll bet that those numbers also have a name. The series
1^1, 2^2, . . . has a rising rate that puts factorials to
shame.

Apparently not, other than n^n, but you can read all about it at https://oeis.org/A000312

To clarify, I was saying it apparently doesn't have a name. I was not disagreeing with your correct claim that it grows much faster than
factorials.

Stirling's approximation is a formula for large factorials. It
consists of raising n/e to the nth power, then multiplying by the
square root of 2 pi n. It's not exact, but it's pretty close for
large numbers, and much easier to calculate than multiplying large
quantities of consecutive numbers.
--
Keith F. Lynch - http://keithlynch.net/
Please see http://keithlynch.net/email.html before emailing me.
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From Newsgroup: rec.arts.sf.fandom

On 9/26/25 9:09 PM, Keith F. Lynch wrote:

Joy Beeson <jbeeson@invalid.net.invalid> wrote:

While counting my calf raises tonight, it struck me that two to the
third was followed immediately by three to the second.

I wonder what pairs like this are called?

Perfect powers. Catalan's Conjecture -- which has since been
proven -- is that 8 and 9 are the only adjacent ones. See https://oeis.org/A001597

Note that 2025 is a perfect power, the first since 1936 and the last
until 2048.

The question was whether such pairs have a name. You're saying that 8
and 9 are the only pair as Joy describes it, so it's likely that kind of
pair doesn't have a name.
--
Gary McGath http://www.mcgath.com
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Gary McGath <garym@mcgath.com> wrote:

The question was whether such pairs have a name. You're saying that
8 and 9 are the only pair as Joy describes it, so it's likely that
kind of pair doesn't have a name.

Apparently so. I checked "Catalan pairs," but that turns out to
be the name of a different concept named for the same 19th century mathematician.

He wasn't actually the first to make the conjecture, just the first to
widely publicize it. It's also called Mihailescu's theorem. The old
name is mostly still used since the proof is so new, dating to 2002.

Ironically, it was proven just for squares and cubes by Gersonides in
1343. So someone partially solved it half a millennium before Catalan
even conjectured it.

But it was far from the oldest math problem. Whether odd perfect
numbers exist has been speculated about for at least 2400 years. (A
perfect number is a number equal to the sum of its proper divisors.)
52 such numbers are known, most of them discovered by exhaustive
computer searches in our lifetime, but all of them are even. Odd
perfect numbers have neither been found nor proven impossible -- which
is odd indeed.

It's good to see discussion here on such an interesting topic.
Indeed, there's a simple proof that every positive integer is
interesting: If any integers were not interesting, there must
be a first one, and that would make that first one interesting,
which would be a contradiction.

This proof can't be generalized to real numbers. Most real numbers
are not only not interesting, but are actually impossible to compute,
specify, or even think about. They have a rather ghostly existence.
It's been controversially suggested that only *computable* numbers
actually exist. But that's obviously false. The Chaitin constants
are not computable but are well-defined and have definite values.
--
Keith F. Lynch - http://keithlynch.net/
Please see http://keithlynch.net/email.html before emailing me.
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