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All Integers are Interesting (with Proof)
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Phillip L. Emerson
science forum beginner


Joined: 28 Apr 2005
Posts: 5

PostPosted: Thu Apr 28, 2005 3:19 pm    Post subject: Re: help - maximum likelihood estimator Reply with quote

"Rusty" <john@hudson8889.fsnet.co.uk> wrote in message
news:d3tncc$pk7$1@news7.svr.pol.co.uk...
Quote:



The interval w can be fitted anywhere that includes Xmax and
Xmin.samples.
The likelihood must be constant for all such locations. So there is no
maximum likelihood point for the mean, only a uniform likelihood for
this
range of locations.



A curious circumstance. I suppose a piling up of observations in a
small
region between the two extremes would cast doubt on the assumptions,
as would any pair of observations farther apart than the "known range.
It is logical, but strange, that the likelihhood outside the observed
extremes
is zero.


Not sure I understand what follow the last sentence exactly


Yeah, I see what you mean. It is not true. Sorry.
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Pavel314
science forum addict


Joined: 29 Apr 2005
Posts: 78

PostPosted: Thu Apr 28, 2005 3:19 pm    Post subject: Re: Measuring Spread of Combinations Reply with quote

"Reef Fish" <Large_Nassau_Grouper@Yahoo.com> wrote in message
news:1114361157.935875.229730@z14g2000cwz.googlegroups.com...

Quote:
So, what's so interesting about numbers like 714 or 715?
Well, on April 8, 1974, Hank Aaron's 715th home run eclipsed
Babe Ruth's 1935 record of 714. Pomerance, a Georgia Asst Prof,
also noticed that 714x715= 2x3x5x7x11x13x17, the product of the
first 7 primes!


714 is one of my favorite numbers. Besides being the number of home runs hit
by Babe Ruth, it was also the badge number of Joe Friday on the original
"Dragnet". And 7/14 in American date notation is July 14, Bastille Day!

Thanks for the 714 X 715 information; I never noticed that before.

There's no such thing as an uninteresting number.


Paul
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Reef Fish
science forum Guru Wannabe


Joined: 28 Apr 2005
Posts: 200

PostPosted: Thu Apr 28, 2005 3:19 pm    Post subject: All Integers are Interesting (with Proof) Reply with quote

Pavel314 wrote:
Quote:
"Reef Fish" <Large_Nassau_Grouper@Yahoo.com> wrote in message
news:1114361157.935875.229730@z14g2000cwz.googlegroups.com...

So, what's so interesting about numbers like 714 or 715?
Well, on April 8, 1974, Hank Aaron's 715th home run eclipsed
Babe Ruth's 1935 record of 714. Pomerance, a Georgia Asst Prof,
also noticed that 714x715= 2x3x5x7x11x13x17, the product of the
first 7 primes!


714 is one of my favorite numbers. Besides being the number of home
runs hit
by Babe Ruth, it was also the badge number of Joe Friday on the
original
"Dragnet". And 7/14 in American date notation is July 14, Bastille
Day!

Thanks for the 714 X 715 information; I never noticed that before.

I am glad I stumbled into interest and your interesting number.

Hardy (a great mathematician by his own record) rated himself 20 on a
scale of 1 to 100, and the self-taught Indian postal clerk Ramamujan,
100.
When Ramanujan, was lying in a nursing-home with a mysterious illness,
Hardy called to see him. Finding making conversation difficult, Hardy
observed lamely that he had come in Taxi no. 1729, which he considered
to be an uninteresting number. Ramanujan immediately perked up, and
said that on the contrary, 1729 was a very interesting number: It was
the SMALLEST positve integer that could be expressed as the sum of two
cubes in two different ways:

1729 = 1^3 + 12^3 = 9^3 + 10^3


Quote:
There's no such thing as an uninteresting number.

Actually, that statement can only be PROVED if the number is an
INTEGER.

Proof:

We proceed by the method of reductio ad absurdum

Suppose the set U of un-interesting integers is non-empty. Then, since

it is a set of integers, there is a least member of U, u, say. u has
the property of being the smallest un-interesting integer, which is
interesting --- a logical contradiction. Thus we have disproved the
hypothesis that U is non-empty, hence empty. The set of un-interesting
integers is empty means there are no un-interesting integers.

QED.

-- Bob.
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Reef Fish
science forum Guru Wannabe


Joined: 28 Apr 2005
Posts: 200

PostPosted: Thu Apr 28, 2005 3:19 pm    Post subject: Re: All Integers are Interesting (with Proof) Reply with quote

Jim Spriggs wrote:
Quote:
Reef Fish wrote:

...

Actually, that statement can only be PROVED if the number is an
INTEGER.

Proof:

We proceed by the method of reductio ad absurdum

Suppose the set U of un-interesting integers is non-empty. Then,
since

You need to say _positive_ integers, else U can be

{-1, -2, -3, -4, ...}

Thank you for your interest in the subject. Wink Yes, U can be any
negative integers too. Nothing in the proof below assumes U or u to
be positive integers only.

If the smallest "un-interesting integer" is -12345679, or any other
negative integer, the proof still holds.

Quote:

it is a set of integers, there is a least member of U, u, say. u
has
the property of being the smallest un-interesting integer, which is
interesting --- a logical contradiction. Thus we have disproved
the
hypothesis that U is non-empty, hence empty. The set of
un-interesting
integers is empty means there are no un-interesting integers.

QED.

-- Bob.

-1 is certainly an interesting integer. The square root of -1 takes
you
from the real domain to the complex domain is just one of the
"interesting"
facts about it. :-)

-25,361,761 is interesting because ... according to the Guiness Book
....

The largest slot machine payout is $39,713,982.25 (25,361,761), won
by
a 25-year-old software engineer (hence -25,361,761 won by the slot
machine <g>) from Los Angeles after putting in $100 (64) in the
Megabucks slot machine at the Excalibur Hotel-Casino (pictured),
Las Vegas, Nevada, USA, on March 21, 2003

-- Bob.
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Stig Holmquist
science forum beginner


Joined: 30 Apr 2005
Posts: 48

PostPosted: Thu Apr 28, 2005 3:19 pm    Post subject: Re: Measuring Spread of Combinations Reply with quote

On Sun, 24 Apr 2005 10:13:30 -0400, "Pavel314" <Pavel314@comcast.net>
wrote:

Quote:
This problem arose from losing the Mega Millions lottery drawing last
Friday. The game is played by picking a set of five numbers from 1 to 52,
plus the Mega Ball number, also from 1 to 52. This problem ignores the Mega
Ball number and concentrates on the non-Mega combination.

I noticed that the winning five non-Mega numbers (23,25,43,46,49) were
"bunched up" in the higher range of possibilities and wondered if there were
some way to measure the spread of five-number picks. I tried two approaches,
statistical and geometric, which gave conflicting results.

STATISTICAL

I used three combinations for this test, took the mean and standard
deviation of each combination, then divided standard deviation by mean to
get a measure of spread around the mean. The first combination was chosen so
that the results would be evenly spread across the possible values; its
ratio is taken as the standard. The second is the winning combination and
the third is a purposely distorted combination. Results were:

(1,14,26,39,52) m=26.40, sd=20.08, ratio= 0.76
(23,25,43,46,49) m=37.20, sd=12.26, ratio=0.33, to standard =43%
(1,15,50,51,52) m=33.80, sd=24.08, ratio=0.71, to standard 94%
Your choise of standard was not realistic. A better set would be
7-16-26-35-45 which ahs a std.dev. of 15.02. The max. std.dev.

for a set of five must be 27.12 and the min. must be 1.58 and thus you
might expect a mean std.dev. of 14.35, but the distribution of s.d is
not symmetrical because the min. will be generated by all sets of
five consequtive numbers, of which there are 47 sets but there are
only two sets yielding 27.12. Also, the distribution is not otherwise
a perfectly normal one. You might consider listing all 5/52 draws
on a spreadsheet and calculate the actual mean std.dev. and you
are likely to find it to be close to 14.5.

Also, it is common to use the sum, not the mean, of the set of five
numbers. It must have an average of 132.5 with a std.dev. of 32.2
which is nearly perfectly normal. Your sum was 185 with a 1.66 s.d.
spread from the mean 132.5..

Bertil
Quote:
This seems to indicate that the spread on the winning numbers was 43% of the
spread of the standard, fairly bunched up, while the third, purposely
distorted combination had 94% of the spread of the standard, hence fairly
widely distributed.

GEOMETRIC

Just out of curiosity, I also found the "hypotenuse" defined by the numbers
of each combination, treating them as a four-dimensional triangle of sorts.
This was done by taking the difference between successive numbers in each
combination, squaring each, summing the squares and taking the square root
of the sum. Results were:

(1,14,26,39,52) hypotenuse= 25.51
(23,25,43,46,49) hypotenuse= 18.60, to standard =73%
(1,15,50,51,52) hypotenuse= 37.72, to standard 148%

Under this system of measure, the winning numbers are still more compact
than the standard, although not as much as under the statistical method.
However, the purposely distorted combination now has a higher spread value
than the standard.


While I trust the statistical results more than the geometric, it seems that
the geometric method has the germ of a meaningful statistic somewhere within
it. Any insights will be appreciated.


Paul



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Reef Fish
science forum Guru Wannabe


Joined: 28 Apr 2005
Posts: 200

PostPosted: Thu Apr 28, 2005 3:19 pm    Post subject: Re: All Integers are Interesting (with Proof) Reply with quote

Reef Fish wrote:
Quote:
Jim Spriggs wrote:
Reef Fish wrote:

...

Actually, that statement can only be PROVED if the number is an
INTEGER.

Proof:

We proceed by the method of reductio ad absurdum

Suppose the set U of un-interesting integers is non-empty. Then,
since

You need to say _positive_ integers, else U can be

{-1, -2, -3, -4, ...}

Thank you for your interest in the subject. Wink Yes, U can be any
negative integers too. Nothing in the proof below assumes U or u to
be positive integers only.

I stand corrected! The proof was indeed valid for positive integers
only, because it assumes there is a SMALLEST element u.

The proof can easily be fixed by amending a 2nd part for NEGATIVE
integers. Let U be the set of all NEGATIVE integers, ... and let u
be the LARGEST of the "un-interesting negative integers" ...

The method of reductio ad absurdum will prove that the set of
"un=interesting negative integers" is an empty set.

Combining the two separate proofs for "positive" and for "negative"
integers, we have proved that there are no un-interesting INTEGERS!

I think I have too much time on my hands ... must go do something
more interesting than this. :=))

-- Bob.


Quote:

If the smallest "un-interesting integer" is -12345679, or any other
negative integer, the proof still holds.


it is a set of integers, there is a least member of U, u, say. u
has
the property of being the smallest un-interesting integer, which
is
interesting --- a logical contradiction. Thus we have disproved
the
hypothesis that U is non-empty, hence empty. The set of
un-interesting
integers is empty means there are no un-interesting integers.

QED.

-- Bob.

-1 is certainly an interesting integer. The square root of -1 takes
you
from the real domain to the complex domain is just one of the
"interesting"
facts about it. :-)

-25,361,761 is interesting because ... according to the Guiness Book
...

The largest slot machine payout is $39,713,982.25 (25,361,761), won
by
a 25-year-old software engineer (hence -25,361,761 won by the slot
machine <g>) from Los Angeles after putting in $100 (64) in the
Megabucks slot machine at the Excalibur Hotel-Casino (pictured),
Las Vegas, Nevada, USA, on March 21, 2003

-- Bob.
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gerard46
science forum beginner


Joined: 24 Mar 2005
Posts: 46

PostPosted: Thu Apr 28, 2005 3:19 pm    Post subject: Re: All Integers are Interesting (with Proof) Reply with quote

| Reef Fish wrote:
|> gerard46 wrote:
-----snipped-----
|>| While we are at it, here are still other interesting facts about the numbers
|>| 714 and 715, besides having the Pomerance property that 714 x 715 is
|>| the product of the first seven primes
|>| 714 + 715 = 1429, which was the year Columbus stumbled onto America.
|>| :^)
|> Before he was even born? _________________________________________Gerard S.

| Yes! That's why he discovered it in his reincarnation in 1492! Smile
|
| Actually, it is now commonly accepted (outside of the USA) that America
| was discovered before Columbus was born!
-----snipped-----

When America was "discovered" wasn't my point, I was pointing out the
error when Christopher stumbled onto America (before he was born?).
______________________________________________________________________Gerard S.
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Carl G.
science forum beginner


Joined: 18 May 2005
Posts: 18

PostPosted: Thu Apr 28, 2005 3:19 pm    Post subject: Re: All Integers are Interesting (with Proof) Reply with quote

"Reef Fish" <Large_Nassau_Grouper@Yahoo.com> wrote in message
news:1114374431.963748.290770@o13g2000cwo.googlegroups.com...

Quote:
There's no such thing as an uninteresting number.

Actually, that statement can only be PROVED if the number is an
INTEGER.

Proof:

We proceed by the method of reductio ad absurdum

Suppose the set U of un-interesting integers is non-empty. Then, since

it is a set of integers, there is a least member of U, u, say. u has
the property of being the smallest un-interesting integer, which is
interesting --- a logical contradiction. Thus we have disproved the
hypothesis that U is non-empty, hence empty. The set of un-interesting
integers is empty means there are no un-interesting integers.

QED.

-- Bob.


Have you considered this classic response to your proof?

Theorem: All integers are boring.

Proof (by contradiction):

Suppose the set of non-boring integers is non-empty, and x is the least
member of this set.

Who cares?

QED. ;-)

Carl G.
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Reef Fish
science forum Guru Wannabe


Joined: 28 Apr 2005
Posts: 200

PostPosted: Thu Apr 28, 2005 3:19 pm    Post subject: Re: Measuring Spread of Combinations Reply with quote

Pavel314 wrote:
Quote:
This problem arose from losing the Mega Millions lottery drawing last

Friday. The game is played by picking a set of five numbers from 1 to
52,


Quote:
I noticed that the winning five non-Mega numbers (23,25,43,46,49)
were
"bunched up" in the higher range of possibilities

There are two completely separate problems here.

1. Probability. If the RANDOM drawing mechanism is indeed random,
then EVERY combination of 5 numbers is equally likely. Nothing
more can be said about (23,25,43,46,49) other than it's probability
is 1/C(52,5), and is as likely as (1,2,3,4,5) or (3,17,29,40,42) or
any combination actually drawn.

2. Statistics. Can you draw some statistical inference based on the
observed numbres whether the generating process is random of not.
NO! Absolutely not! In view of (1), when the generating process
is truly random, every observed quintuple is equally like.

All such "odd" outcomes can suggest is for one to examine whether
the physical process that generated the outcome has flaws in terms
of making them all equally like.

3. Mathematics and Number theory. There is something "interesting"
about every one of the numbers drawn, as some examples of such
were mentioned in the book "The Man Who Loved Only Numbers". In
the observed numbers, 23 and 43 are primes, 25 and 49 are perfect
squares, and 46? 46 can have lots of different interesting
properties. For one, it is the sum of the Fibonnaci numbers
(1,3,8,13,21) <-- I found that just now. <G> It is also the
sum of these 4 PRIME numbers (3+5+7+31), three of which are the
smallest Mersenne primes <-- my discovery this morning too, about
the number 46. Smile 46 is also the sum of 4 perfect numbers
(6+6+6+2Cool. And this could go on and on and on ... that every
integer has many "interesting" properties.

So, what's so interesting about numbers like 714 or 715?
Well, on April 8, 1974, Hank Aaron's 715th home run eclipsed
Babe Ruth's 1935 record of 714. Pomerance, a Georgia Asst Prof,
also noticed that 714x715= 2x3x5x7x11x13x17, the product of the
first 7 primes!

Albert Wilanski was the only one in the world who noticed that
the phone number of his brother-in-law had the peculiar property
the sum of its digits is equal to the sum of its prime factors!
The phone number was 493-7775. 4,937,775 can be expressed as the
product of its prime factors 3x5x5x65837. The sum of the original
digits (4+9+3+7+7+7+5) was 42, so was the sum of the digits in the
prime factors (3+5+5+6+5+8+3+7).

Take another phone number, say 641-1024. Paul Erdos will be able
to tell you immediately that it is 2532 squared. Erdos knows the
square of any 4-digit number, and in his words, "sorry I am getting
old and cannot tell you the cube."


An example of (2) was in the draft lottery used by Uncle Sam. Those
born early in the year noticed many more of them were drafted than
those
born in the second half of the year.

But in the random SEQUENCE, all 365! sequences are equally likely. So,
even if the sequence drawn turned out to be 1,2,3,4,...,365 in perfect
order, it has the SAME probability of being drawn as any other. So,
it's
an exercise in futility if one tries to analyze the observed sequence
to
argue whether the generating process is random or not.

But it DID point to the examination of how the numbers were selected,
and
it pointed to certain noticeable flaws that explained the anamoly of
the
observed sequence.

If one wanted "equal representation" in whatever respect, all one needs
to do is to have a "stratefied random" sample (or sequence).

"What if you threw up a set of wood blocks of letters, and it spelled
Christmas?" That's Jimmie Savage's favorite example about
"randomization".


Your "statistical" and "geometric" analyses of the OBSERVD numbers
sounds
interesting, but so does "numerology", which has no probability or
statistical base or content. Try NUMBER THEORY. You may find very
interesting properties of those numbers. :)

-- Bob.

Quote:
STATISTICAL

I used three combinations for this test, took the mean and standard
deviation of each combination, then divided standard deviation by
mean to
get a measure of spread around the mean. The first combination was
chosen so
that the results would be evenly spread across the possible values;
its
ratio is taken as the standard. The second is the winning combination
and
the third is a purposely distorted combination. Results were:

(1,14,26,39,52) m=26.40, sd=20.08, ratio= 0.76
(23,25,43,46,49) m=37.20, sd=12.26, ratio=0.33, to standard =43%
(1,15,50,51,52) m=33.80, sd=24.08, ratio=0.71, to standard 94%

This seems to indicate that the spread on the winning numbers was 43%
of the
spread of the standard, fairly bunched up, while the third, purposely

distorted combination had 94% of the spread of the standard, hence
fairly
widely distributed.

GEOMETRIC

Just out of curiosity, I also found the "hypotenuse" defined by the
numbers
of each combination, treating them as a four-dimensional triangle of
sorts.
This was done by taking the difference between successive numbers in
each
combination, squaring each, summing the squares and taking the square
root
of the sum. Results were:

(1,14,26,39,52) hypotenuse= 25.51
(23,25,43,46,49) hypotenuse= 18.60, to standard =73%
(1,15,50,51,52) hypotenuse= 37.72, to standard 148%

Under this system of measure, the winning numbers are still more
compact
than the standard, although not as much as under the statistical
method.
However, the purposely distorted combination now has a higher spread
value
than the standard.


While I trust the statistical results more than the geometric, it
seems that
the geometric method has the germ of a meaningful statistic somewhere
within
it. Any insights will be appreciated.


Paul
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Reef Fish
science forum Guru Wannabe


Joined: 28 Apr 2005
Posts: 200

PostPosted: Thu Apr 28, 2005 3:19 pm    Post subject: Re: All Integers are Interesting (with Proof) Reply with quote

Carl G. wrote:
Quote:
"Reef Fish" <Large_Nassau_Grouper@Yahoo.com> wrote in message
news:1114374431.963748.290770@o13g2000cwo.googlegroups.com...

There's no such thing as an uninteresting number.

Actually, that statement can only be PROVED if the number is an
INTEGER.

Proof:

We proceed by the method of reductio ad absurdum

Suppose the set U of un-interesting integers is non-empty. Then,
since

it is a set of integers, there is a least member of U, u, say. u
has
the property of being the smallest un-interesting integer, which is
interesting --- a logical contradiction. Thus we have disproved
the
hypothesis that U is non-empty, hence empty. The set of
un-interesting
integers is empty means there are no un-interesting integers.

QED.

-- Bob.


Have you considered this classic response to your proof?

Theorem: All integers are boring.

Not really.

Quote:

Proof (by contradiction):

Suppose the set of non-boring integers is non-empty, and x is the
least
member of this set.

Who cares?

Apparently Carl G. did.

Quote:

QED. Wink

You need to patch your proof because you haven't proved that YOUR
x (the least number of the set) exists.

Quote:

Carl G.

Perhaps this interests you even less, though it CAN be proven by
reductio ad absurdum because there are only 8700 threads in all
newsgroups posted by Carl G (according to Google search) and most
of them are not even by YOU:

Theorem: All postings in newsgroups authored by Carl G are boring.

Buenos tardes,

la Poisson.
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nooneimportant
science forum beginner


Joined: 28 Apr 2005
Posts: 1

PostPosted: Thu Apr 28, 2005 3:19 pm    Post subject: Re: All Integers are Interesting (with Proof) Reply with quote

You guys are out there.
I'm just happy when I can help my 10 year old with a
I did find this newsgroup interesting. Just reading the stuff you genius
people write is amazing.
word problem.
"Reef Fish" <Large_Nassau_Grouper@Yahoo.com> wrote in message
news:1114405436.084035.284560@o13g2000cwo.googlegroups.com...
Quote:

Bob Pease wrote:
"Reef Fish" <Large_Nassau_Grouper@Yahoo.com> wrote in message
news:1114374431.963748.290770@o13g2000cwo.googlegroups.com...

Gigantic snip for brevity


Suppose that Dustin Hoffman found nothing interesting about anything
except
1,2,
Someone points out that the number 3 is the first uninteresting
number,
then.
So Dustin admits this and groups all numbers this way.
so if you name any number whatever, it is now interesting to HIM
because of
its residue mod 2.

This joke is almost as bad as the joke of Cauchy's dog leaving his
residue at the pole.


( reducing Dustin's world to binary)

Another oldie:
There are 10 kinds of people in the world:
those who understand binary and those who don't.


Even *I* think this post is getting odd!!

On even and odd, Goldback's conjecture that all EVEN numbers greater
than 2 can be expressed as the sum of two (ODD of course) primes
is still up for grabs.

The conjecture had been verified to all even numbers up to 100 million,
but has not been proved to be universally true for all even numbers.


Bob Pease

A final interesting number for today. Mersennes conjectured that
2^67 - 1 was a prime, after it was found that 2^p -1 were prime for
p = 2,3,5,7 resulted in primes, but failed when p =11.

The conjecture stood for 250 years unchallenged until Frank Nelcon Cole
gave a "talk" at the American Mathematics Society in 1903. Cole, a man
of few words, did not say a single during the entire presentation.
He went to the blackboard. raised 2 to the 67th power and subtracted 1
to get 147,573,952,589,676,412,927.

Without a word, he moved to the clear side of the blackboard and
multiply out long hand,

195,707,721 x 761,838,257,287

which was 147,573,952,589,676,412,927.

That poor sap had more time on his hand than any of us! Smile But
he certainly made the Mersenne number 147,573,952,589,676,412,927
interesting.

-- Bob.

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Reef Fish
science forum Guru Wannabe


Joined: 28 Apr 2005
Posts: 200

PostPosted: Thu Apr 28, 2005 3:19 pm    Post subject: Re: All Integers are Interesting (with Proof) Reply with quote

gerard46 wrote:
Quote:
| Reef Fish wrote:
|> gerard46 wrote:
-----snipped-----
|>| While we are at it, here are still other interesting facts about
the numbers
|>| 714 and 715, besides having the Pomerance property that 714 x 715
is
|>| the product of the first seven primes
|>| 714 + 715 = 1429, which was the year Columbus stumbled onto
America.
|>| :^)
|> Before he was even born?
_________________________________________Gerard S.

| Yes! That's why he discovered it in his reincarnation in 1492!
Smile


That was my JOKE. See the SMILEY Smile ? Columbus discovered it in
his previous life. Now re-read the preceding sentence.


Quote:
|
| Actually, it is now commonly accepted (outside of the USA) that
America
| was discovered before Columbus was born!
-----snipped-----

When America was "discovered" wasn't my point, I was pointing out the
error when Christopher stumbled onto America (before he was born?).

______________________________________________________________________Gerard

S.

I know! That's why I wrote also:

RF> LOL! You caught my hidden Dyslexic Devil!

Let me explain. A dyslexic person is one who is prone to having
problems
with left-right distinction and tends to permute letters or numerals.

So, my Dyslexic Devil wrote "1429" instead of "1492" transposing the
last two digits.

See if you get this dyslexlic joke:

"The Dyslexic Agnostic stayed up nights wondering if there is a Dog."


I rather suspect you're not THAT humor challenged, but rather English
is not your native or commonly used language, and hence the meanings
of those words like "re-incarnation" and "dyslexic" did not register
in intended jokes.

English is not my native language either. But I manage, with or
without the help of my Keyboard Devil, Dyslexic Devil, and other
Devils.

Did you know that the Devil number 666 is VERY VERY Interesting?

http://users.aol.com/s6sj7gt/mike666.htm

-- Bob.
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Mike Terry
science forum Guru Wannabe


Joined: 02 May 2005
Posts: 137

PostPosted: Thu Apr 28, 2005 3:19 pm    Post subject: Re: All Integers are Interesting (with Proof) Reply with quote

"Frank J. Lhota" <NOSPAM.Lhota.adarose@verizon.net> wrote in message
news:GyRbe.7643$WX.1703@trndny01...
Quote:
"Reef Fish" <Large_Nassau_Grouper@Yahoo.com> wrote in message
news:1114374431.963748.290770@o13g2000cwo.googlegroups.com...
Actually, that statement can only be PROVED if the number is an
INTEGER.

Not necessarily. Using the Axiom of Choice, every set can be well ordered,
that is, every set S has a linear ordering such that every non-empty
subset
of S has a minimal member. Your proof can be modified to show that any set
that can be well ordered has no uninteresting members.

This only works if the well ordering is a "particular" well ordering that is
interesting in some way... Smile
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Carl G.
science forum beginner


Joined: 18 May 2005
Posts: 18

PostPosted: Thu Apr 28, 2005 3:19 pm    Post subject: Re: All Integers are Interesting (with Proof) Reply with quote

"Reef Fish" <Large_Nassau_Grouper@Yahoo.com> wrote in message
news:1114650523.214224.28890@o13g2000cwo.googlegroups.com...
Quote:

Carl G. wrote:
"Reef Fish" <Large_Nassau_Grouper@Yahoo.com> wrote in message
news:1114374431.963748.290770@o13g2000cwo.googlegroups.com...

There's no such thing as an uninteresting number.

Actually, that statement can only be PROVED if the number is an
INTEGER.

Proof:

We proceed by the method of reductio ad absurdum

Suppose the set U of un-interesting integers is non-empty. Then,
since

it is a set of integers, there is a least member of U, u, say. u
has
the property of being the smallest un-interesting integer, which is
interesting --- a logical contradiction. Thus we have disproved
the
hypothesis that U is non-empty, hence empty. The set of
un-interesting
integers is empty means there are no un-interesting integers.

QED.

-- Bob.


Have you considered this classic response to your proof?

Theorem: All integers are boring.

Not really.


Proof (by contradiction):

Suppose the set of non-boring integers is non-empty, and x is the
least
member of this set.

Who cares?

Apparently Carl G. did.


QED. ;-)

You need to patch your proof because you haven't proved that YOUR
x (the least number of the set) exists.


Carl G.

Perhaps this interests you even less, though it CAN be proven by
reductio ad absurdum because there are only 8700 threads in all
newsgroups posted by Carl G (according to Google search) and most
of them are not even by YOU:

Theorem: All postings in newsgroups authored by Carl G are boring.

Buenos tardes,

la Poisson.

I didn't create this "proof" (or "spoof"). It has been around for many
years (I found it in a list of mathematics jokes). I actually find some
integers interesting (for example: 7101001000).

I agree that many (if not most) people in the world would find my posts
boring. The exact number might be interesting to some people, but not to
me. ;-)

Carl G.
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Yaroslav Bulatov
science forum beginner


Joined: 28 Apr 2005
Posts: 23

PostPosted: Thu Apr 28, 2005 6:58 pm    Post subject: Re: Measuring Spread of Combinations Reply with quote

You can determine how random a sequence is by using an appropriate test
for randomness of sequences. For instance if the sequence you got had
only 0's or 1's, you could use the runs test. Observing a sequence
1111111 would give strong evidence against underlying generating
process being random.

In your case there are no regular runs, but there is a run with step 3
(43,46,49). So if you devised a runs test which used runs of this kind
before seeing the data, and then saw the sequence, you'd have evidence
against the process being random.

An important issue is to devise the test before seeing the data. If you
pick runs test because your sequence has runs, or pick some test on the
histogram because distribution of digits seems skewed, the type-1 and
type-2 errors associated with the hypothesis test are no longer valid.
This means that you don't know whether results of those tests carry any
statistical significance.
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Google

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