• ## RE: [PrimeNumbers] Recursive and Pure Primes

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• Carl Sagan also included 1 as prime among the billions and billions of stars in his Cosmos. I think Martin Gardner was also another Rebel.
Message 1 of 17 , Mar 29 12:48 PM
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Carl Sagan also included 1 as prime among the "billions and billions of stars" in his
Cosmos. I think Martin Gardner was also another "Rebel."

http://primes.utm.edu/curios/page.php/1.html

Cino

From: caldwell@...
Date: Sun, 29 Mar 2009 08:04:13 -0500
Subject: RE: [PrimeNumbers] Recursive and Pure Primes

> By the way, I read here
(http://www.geocities.com/primefan/Prime1ProCon.html)
> and elsewhere that 1 was prime until 1899 when it was changed "to
preserve the
> uniqueness in the Fundamental Theorem of Arithmetics". How true is
that? and
> fundamentally shouldn't 1 be the prime of primes :)

It is about 7% true. When the early writers defined primes (Euclid et
al.), 1 was not even considered a number--it was the unit and numbers
were multiples of it. So it was not prime.

However after 1 became a number (see most any math history book), many
did consider it a prime--including many true number theorists. (And
long after 1899.)

However, as the study of number fields matured it was obvious we needed
to be more careful and 1 was was returned to its rightful spot.

See, for example, http://primes.utm.edu/notes/faq/one.html

[Non-text portions of this message have been removed]
• ... The number 1 is an extinct prime since it was once thought to be prime by many and now is no longer considered to be prime. [Hilliard] ...makes me
Message 2 of 17 , Apr 5, 2009
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--- In primenumbers@yahoogroups.com, cino hilliard wrote:

"
The number 1 is an "extinct" prime since it was once thought to be prime by many and now is no longer considered to be prime. [Hilliard]
"

...makes me think (with some melancholy) of the (ex-)planet Pluto...

Maximilian

PS: (to "The Editors") The given names of Riemann are both spelled wrong on that page, they are not "George Bernard" but "Georg (Friedrich) Bernhard".
• Im reposting this, as i realize i sent it to a person, not the group :P years ago, i had a teacher that chose to simplify this problem by comparing it to the
Message 3 of 17 , Apr 6, 2009
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Im reposting this, as i realize i sent it to a person, not the group :P

years ago, i had a teacher that chose to simplify this problem by
comparing it to the number 0.

he gave us a simple problem where a/a = 1 and asked what the solution
was. Of coarse, everyone stated that a was any number, real or
imaginary.

it was the wrong answer.. why? As the teacher said, any answer must
preserve all math rules, not just ones that the mathematician chooses
to follow.

In the above equation, 0 can not be an answer because you are not
allowed to divide by 0. 0/0 does not equal 1

what does this have to do with 1 not being a prime?

assume a^2 = k. k is not a prime because it factors into a * a * 1.

since 1 is 1^2, 1 must not be a prime.

in other words, claiming 1 as a prime ignores the fact that a number
that is a perfect square by definition can not be a prime.

this does bring up an interesting question though.. could the number
-1 be concidered the ONLY negative prime.. since it is -1 * 1?

if all numbers can factor into primes, numbers like -4 can factor into 2*2*-1

Yes i know that the definition of primes state only positive numbers,
but besides that, does -1 pass prime tests?
• And if you look at the numbers from 1 to 10 (number of fingers God gane us) then there is VERY CLEAR mirror symmetry between primes and composites if 1
Message 4 of 17 , Apr 6, 2009
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And if you look at the numbers from 1 to 10 (number of fingers God gane us) then there is VERY CLEAR mirror symmetry between primes and composites if 1 is considered a prime, as follows:

1 2 3 _ 5 _ 7 _ _ _     versus     _ _ _ 4 _ 6 _ 9 8 10

I just discovered this two days ago :) It may have been spotted before but for me this is yet another argument in favor of 1 becoming a prime and the best of primes (purest prime I hope :)

<prime numbers are God's signature>

________________________________
From: cino hilliard <hillcino368@...>
Sent: Monday, March 30, 2009 3:48:01 AM
Subject: RE: [PrimeNumbers] Recursive and Pure Primes

Carl Sagan also included 1 as prime among the "billions and billions of stars" in his
Cosmos. I think Martin Gardner was also another "Rebel."

http://primes. utm.edu/curios/ page.php/ 1.html

Cino

From: caldwell@utm. edu
Date: Sun, 29 Mar 2009 08:04:13 -0500
Subject: RE: [PrimeNumbers] Recursive and Pure Primes

> By the way, I read here
(http://www.geocitie s.com/primefan/ Prime1ProCon. html)
> and elsewhere that 1 was prime until 1899 when it was changed "to
preserve the
> uniqueness in the Fundamental Theorem of Arithmetics" . How true is
that? and
> fundamentally shouldn't 1 be the prime of primes :)

It is about 7% true. When the early writers defined primes (Euclid et
al.), 1 was not even considered a number--it was the unit and numbers
were multiples of it. So it was not prime.

However after 1 became a number (see most any math history book), many
did consider it a prime--including many true number theorists. (And
long after 1899.)

However, as the study of number fields matured it was obvious we needed
to be more careful and 1 was was returned to its rightful spot.

See, for example, http://primes. utm.edu/notes/ faq/one.html

[Non-text portions of this message have been removed]

[Non-text portions of this message have been removed]
• Dear All,   I have found that both 123571113171923 and 231917131175321 are primes with 23 being the number of human chromosomes pairs. I hope this would add
Message 5 of 17 , Apr 7, 2009
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Dear All,

I have found that both 123571113171923 and 231917131175321 are primes with 23 being the number of human chromosomes pairs. I hope this would add some weight to the argument for reconsidering 1 as a prime.

This is in addition to my earlier post of the mirror symmetry of primes versus composites between 1 and 10 (human fingers) as 1 2 3 _ 5 _ 7 _ _ _  and   _ _ _ 4 _ 6 _ 8 9 10.

I am sorry if this subject has been discussed before I joined the group but I really think we need a proper debate (not because the definition says so) for and against 1 being a prime.

The most convincing argument against 1 being a prime is the zeta function’s two equivalent series of natural number sum and prime number product that starts at 2 not 1.

Sum [1 / (n^s)]  =  Product [1 / (1 - (1 / p^s)]

Does anyone sympathize with me :)

<prime numbers are God's signature>

[Non-text portions of this message have been removed]
• Hello, My plausible definition why 1 is not a prime. Would be 1 a prime , so is 1 the only prime. Every number have the primefactor 1. So it is a confliction
Message 6 of 17 , Apr 8, 2009
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Hello,
My plausible definition why 1 is not a prime.

Would be 1 a prime , so is 1 the only prime. Every number have the primefactor 1.
So it is a confliction that the numbers 2,3,5 simultaneously prime and not.

I have create another consideration.

We splitt the natural numbers in groups of dimensions.

D0: 1
D1: 2,3,5,7,11... all primes
D2: 4,6,9,10,14...all semi numbers
D3: 8,12,18,...
D4: 16..
.
.
.

So the primes are a group with the most numbers :-)

best wishes

Norman

--- Ali Adams <alipoland@...> schrieb am Di, 7.4.2009:
Betreff: [PrimeNumbers] Prime 23 points to Prime 1
CC: pureprimes@...
Datum: Dienstag, 7. April 2009, 17:59

Dear All,

I have found that both 123571113171923 and 231917131175321 are primes with 23 being the number of human chromosomes pairs. I hope this would add some weight to the argument for reconsidering  1 as a prime.

This is in addition to my earlier post of the mirror symmetry of primes versus composites between 1 and 10 (human fingers) as 1 2 3 _ 5 _ 7 _ _ _  and   _ _ _ 4 _ 6 _ 8 9 10.

I am sorry if this subject has been discussed before I joined the group but I really think we need a proper debate (not because the definition says so) for and against 1 being a prime.

The most convincing argument against 1 being a prime is the zeta function’s two equivalent series of natural number sum and prime number product that starts at 2 not 1.

Sum [1 / (n^s)]  =  Product [1 / (1 - (1 / p^s)]

Does anyone sympathize with me :)

<prime numbers are God's signature>

[Non-text portions of this message have been removed]

[Non-text portions of this message have been removed]
• Excellent groping idea Norman :) May be the other groups D3, D4, ... are hard to find any distribution patterns to them. May be by studying these groups we
Message 7 of 17 , Apr 8, 2009
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Excellent groping idea Norman :)

May be the other groups D3, D4, ... are hard to find any distribution patterns to them.

May be by studying these groups we can find new hint to the distribution of pri,es.

Thanks a lot.
<prime numbers are God's signature>

________________________________
From: Norman Luhn <nluhn@...>
Sent: Thursday, April 9, 2009 12:23:53 AM
Subject: AW: [PrimeNumbers] Prime 23 points to Prime 1

Hello,
My plausible definition why 1 is not a prime.

Would be 1 a prime , so is 1 the only prime. Every number have the primefactor 1.
So it is a confliction that the numbers 2,3,5 simultaneously prime and not.

I have create another consideration.

We splitt the natural numbers in groups of dimensions.

D0: 1
D1: 2,3,5,7,11.. . all primes
D2: 4,6,9,10,14. ..all semi numbers
D3: 8,12,18,...
D4: 16..
.
.
.

So the primes are a group with the most numbers :-)

best wishes

Norman

--- Ali Adams <alipoland@yahoo. com> schrieb am Di, 7.4.2009:
Betreff: [PrimeNumbers] Prime 23 points to Prime 1
CC: pureprimes@yahoo. com
Datum: Dienstag, 7. April 2009, 17:59

Dear All,

I have found that both 123571113171923 and 231917131175321 are primes with 23 being the number of human chromosomes pairs. I hope this would add some weight to the argument for reconsidering  1 as a prime.

This is in addition to my earlier post of the mirror symmetry of primes versus composites between 1 and 10 (human fingers) as 1 2 3 _ 5 _ 7 _ _ _  and   _ _ _ 4 _ 6 _ 8 9 10.

I am sorry if this subject has been discussed before I joined the group but I really think we need a proper debate (not because the definition says so) for and against 1 being a prime.

The most convincing argument against 1 being a prime is the zeta function’s two equivalent series of natural number sum and prime number product that starts at 2 not 1.

Sum [1 / (n^s)]  =  Product [1 / (1 - (1 / p^s)]

Does anyone sympathize with me :)

<prime numbers are God's signature>

[Non-text portions of this message have been removed]

[Non-text portions of this message have been removed]

[Non-text portions of this message have been removed]
• Excellent groping idea Norman :) Thank you :-)   ...   ... Some days ago, I had the perception: The primes are only a result to put of one on top od the
Message 8 of 17 , Apr 8, 2009
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Excellent groping idea Norman :)

Thank you :-)

>May be the other groups D3, D4, ... are hard to find any distribution >patterns to them.

>May be by studying these groups we can find new hint to the distribution of >primes.

Some days ago, I had the perception:
The primes are only a result "to put of one on top od the other" of
infinit symmetric rasters and (sorry) the "Sieve of Eratosthenes " works NOT !!! I think, I the first person of the world with this opinion.

( The rasters are 2,4,6,8,10 ..., 3,6,9,12,.... 5,10,15,... on so on )

The reason why the sieve works not is:
I can not cancel once more an element in a list. When I cancel the number 6 ( a factor is 2 ) then I can not cancel the 6 for factor 3.
Number 6 is not more in list for factor 3 ( only on paper ). Really
you cancel the number 9,15,21, .. on list for factor 3 and for 5
is it 25,35,55,...

We calculate the primes up to 1 billion and take factor 10007, so we have also calculate 2*10007,3*10007, 5*10007,..and so on ,to get the really free elements on list (It don't to leap to the eyes that number 100'140'049 is the first we can cancel from list :-)

Sorry for my confuse english )

Norman

• ... yes you can, in some sense: assign a score to each number, starting with score(1)=1 and adding the score of each number to all of its multiples. (Other
Message 9 of 17 , Apr 8, 2009
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> ( The rasters are 2,4,6,8,10 ..., 3,6,9,12,.... 5,10,15,... on so on )
>
> The reason why the sieve works not is:
> I can not cancel once more an element in a list.

yes you can, in some sense: assign a score to each number,
starting with score(1)=1 and adding the score of each number to all of its multiples. (Other variations are possible: increase the score of each multiple by 1, ...).
Then primes are those with score 2.

M.
• ... No, U(Z) = { -1, 1 } are the units (= invertible elements) of the ring Z and therefore by definition not irreducible elements. However, { -2, -3, -5, ... }
Message 10 of 17 , Apr 15, 2009
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> this does bring up an interesting question though.. could the number
> -1 be concidered the ONLY negative prime.. since it is -1 * 1?

No, U(Z) = { -1, 1 } are the units (= invertible elements) of the ring Z and therefore by definition not irreducible elements.

However, { -2, -3, -5, ... } could as well be called primes, since irreducible elements are only defined up to a unit.

It is somehow arbitrary that we prefer positive representatives in Z, in the same way as we sometimes prefer "unitary" polynomials (leading coefficient = 1) when we speak of irreducible factors in R[Z] (where the units are all elements of R*).
But one could as well require the trailing coefficient to be = 1. Or any other convention.

In the general situation, we don't have a way to prefer one representative among (the equivalence class of) all associated irreducible elements.
(a,b are associated iff a=ub with a unit u)

> if all numbers can factor into primes, numbers like -4 can factor into 2*2*-1

In general an element factors into irreducible elements which are defined up to a unit.
Just like 2x^2-2 = (x-1)(2x+2) = (2x-2)(x+1)
There is no reason to prefer one of these two factorizations.

In this example you might say the factor 2 has to be separated, too. But this is not an irreducible polynomial. Just like -1 is not a prime number.

> Yes i know that the definition of primes state only positive numbers,
> but besides that, does -1 pass prime tests?

This certainly depends on the implementation....
• ... On Planet Zog the sloths are bipeds, all having the same number of toes, distributed between their two feet, with each foot having at least one toe. All
Message 11 of 17 , Apr 15, 2009
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Ali Poland <alipoland@...> wrote:

> number of fingers God gave us

On Planet Zog the sloths are bipeds, all having the same
number of toes, distributed between their two feet,
with each foot having at least one toe. All possible
sub-species occur. Each sloth has a composite number
of toes on one foot, but not on the other.

Prove that each sloth has 11 toes.

PS: Sloth theologians on Zog have tried to read
significance into the uniqueness of this solution,
non-primality of units.

David (per proxy J. Swift)
• ... In reciprocity theorems, the associate which is often taken to be the primary prime is one with a particular modular relation, and not the one with a
Message 12 of 17 , Apr 16, 2009
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--- On Wed, 4/15/09, Maximilian Hasler <maximilian.hasler@...> wrote:
> > this does bring up an interesting question though..
> could the number
> > -1 be concidered the ONLY negative prime.. since it is
> -1 * 1?
>
> No, U(Z) = { -1, 1 } are the units (= invertible elements)
> of the ring Z and therefore by definition not irreducible
> elements.
>
> However, { -2, -3, -5, ... } could as well be called
> primes, since irreducible elements are only defined up to a
> unit.
>
> It is somehow arbitrary that we prefer positive
> representatives in Z, in the same way as we sometimes prefer
> "unitary" polynomials (leading coefficient = 1) when we
> speak of irreducible factors in R[Z] (where the units are
> all elements of R*).
> But one could as well require the trailing coefficient to
> be = 1. Or any other convention.
>
> In the general situation, we don't have a way to prefer one
> representative among (the equivalence class of) all
> associated irreducible elements.
> (a,b are associated iff a=ub with a unit u)

In reciprocity theorems, the associate which is often taken to be the 'primary prime' is one with a particular modular relation, and not the one with a particular sign. So if looking at quadratic reciprocity, the flattened list of primes might be { 2, -3, 5, -7, -11, 13 ... } (chosen to never be 3 modulo 4). For cubic reciprocity, you'll be looking at Eisenstein primes, and, being 2 dimensional, the pattern isn't quite so obvious, but almost simple to formulate as a modular relation between the real and non-real components.

Phil
• ... Clearly each sloth has at least 11 toes, since no smaller number will do. Kermit sent me, off-line, the germ of the proof that each sloth has *precisely*
Message 13 of 17 , Apr 16, 2009
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> On Planet Zog the sloths are bipeds, all having the same
> number of toes, distributed between their two feet,
> with each foot having at least one toe. All possible
> sub-species occur. Each sloth has a composite number
> of toes on one foot, but not on the other.

Clearly each sloth has at least 11 toes,
since no smaller number will do.

Kermit sent me, off-line, the germ of the
proof that each sloth has *precisely* 11 toes.

Here is my own proof.

Let N be the number of toes of each sloth.

1) N is odd, else a sub-species of
sloth would have N/2 toes on each foot,
which is impossible, since no number
may be both composite and non-composite.

2) We know that N - 2 is composite,
since 2 is prime. Hence N > 9.

3) Since N > 9, we require that
N - 4, N - 6, N - 8 are all prime.
But one those 3 numbers must be divisible by 3.
The only prime divisible by 3 is 3.
Hence N - 8 = 3.
So N = 11 and we are done.

Comment: My anti-theological bias derives from

http://www.gutenberg.org/dirs/etext97/gltrv10.txt

"Which two mighty powers have, as I was going to tell you,
been engaged in a most obstinate war for six-and-thirty
moons past. It began upon the following occasion. It is
allowed on all hands, that the primitive way of breaking
eggs, before we eat them, was upon the larger end; but his
present majesty's grandfather, while he was a boy, going to
eat an egg, and breaking it according to the ancient
practice, happened to cut one of his fingers. Whereupon the
emperor his father published an edict, commanding all his
subjects, upon great penalties, to break the smaller end of
their eggs. The people so highly resented this law, that
our histories tell us, there have been six rebellions raised
on that account; wherein one emperor lost his life, and
another his crown. These civil commotions were constantly
fomented by the monarchs of Blefuscu; and when they were
quelled, the exiles always fled for refuge to that empire.
It is computed that eleven thousand persons have at several
times suffered death, rather than submit to break their eggs
at the smaller end. Many hundred large volumes have been
published upon this controversy: but the books of the
Big-endians have been long forbidden, and the whole party
rendered incapable by law of holding employments."

Fortunately, debate on the non-primality of units
has not, so far as I am aware, led to any fatality.

David (per proxy J. Swift)
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