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*unique* prime factorizations; the fundamental theorem of arithmetic
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DGoncz@aol.com
science forum Guru Wannabe


Joined: 25 Oct 2005
Posts: 122

PostPosted: Sun Jul 16, 2006 9:53 am    Post subject: *unique* prime factorizations; the fundamental theorem of arithmetic Reply with quote

1) Has the fundamental theorem of arithmetic got a proof?

I know that for every possible combination of e_1, e_2, etc. and every
n the sum

sum with (i = 1 to n) of (p1^e1, p2^e2, ... , pn^en) is unique,
but for given m,

is the set of primes <= m defined as

{p1, p2, ... , pn} | ( p1 < p2 < ... pn <= m ) and (sum above is
unique) equal to

the unique set of primes as we know them, {2, 3, ... pn <=m},
or are there other sets that have this property?

We can call the infinte set of primes P_oo = {2,3,5,7...} or P = {2(1),
3(1), 5(1), 7(1), ...}.

Now, it seems fhat for every x > 0, a unique set X exists which is the
multiset prime factorization of x. Of course for every prime p the
corresponding set P has one member with multiplicity one.

2) Is the UPF function from {x} to {X} a function? Is it one-to-one? Is
it onto?

3) What are the restrictions on operators allowing translation of
arbitrary statements involving positive integer variables x, y, z, etc.
into logically equivalent statements (propositions) involving
corresponding UPF multisets X, Y, Z, etc?

4) Is the (we'll call it) "descending translation" from operators on
x,y,z to operators on X,Y,Z in general use by number theorists?

For instance,
gcd(x,y) = (x,y) = X /\ Y
lcm(x,y) = [x,y] = X \/ Y
(x,y) * [x,y] = x*y = X + Y adding corresponding multiplicites

so it seems we should be able to write a descending translation table:

gcd --- intersection
lcm --- union
---precendence break
sum --- some thing
difference --- some other thing
---precedence break
product --- + (multiset sum)
quotient --- - (multiset difference)
---precedence break
exponentiation --- * (defined?)
some other thing --- /
---precedencde break
some other thing --- ^ etc.

5) Is there software like MACSYMA or Mathcad that will let you work
with both representations in the same document? Softwre that
facilitates writing "both ways at the same time?"

I'm off to Mathworld and Wikipedia to find out about this.

Doug Goncz
Replikon Research
Seven Corners, VA 22044-0394
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DGoncz@aol.com
science forum Guru Wannabe


Joined: 25 Oct 2005
Posts: 122

PostPosted: Sun Jul 16, 2006 10:09 am    Post subject: Re: *unique* prime factorizations; the fundamental theorem of arithmetic Reply with quote

OK, even Wikipedia has the fundamental theorem, so there's much to read
on this. However, prime signatures are interesting. Sorted in
descending order. Why?

I didn't see the "other possible" prime signature in use:

25 = 5^2 ---> {0,0,2}
18 = 2*3^2 ---> {1,2}

*These* prime signatures are unique and correspond to the integers
one-to-one, so any restrictions on the "descending translation" table
below relating to mutisets can be recast into conventional sets, and
the set P_x the set of primes less than or equal to x.

Doug

Doug Goncz wrote:
Quote:
1) Has the fundamental theorem of arithmetic got a proof?

I know that for every possible combination of e_1, e_2, etc. and every
n the sum

sum with (i = 1 to n) of (p1^e1, p2^e2, ... , pn^en) is unique,
but for given m,

is the set of primes <= m defined as

{p1, p2, ... , pn} | ( p1 < p2 < ... pn <= m ) and (sum above is
unique) equal to

the unique set of primes as we know them, {2, 3, ... pn <=m},
or are there other sets that have this property?

We can call the infinte set of primes P_oo = {2,3,5,7...} or P = {2(1),
3(1), 5(1), 7(1), ...}.

Now, it seems fhat for every x > 0, a unique set X exists which is the
multiset prime factorization of x. Of course for every prime p the
corresponding set P has one member with multiplicity one.

2) Is the UPF function from {x} to {X} a function? Is it one-to-one? Is
it onto?

3) What are the restrictions on operators allowing translation of
arbitrary statements involving positive integer variables x, y, z, etc.
into logically equivalent statements (propositions) involving
corresponding UPF multisets X, Y, Z, etc?

4) Is the (we'll call it) "descending translation" from operators on
x,y,z to operators on X,Y,Z in general use by number theorists?

For instance,
gcd(x,y) = (x,y) = X /\ Y
lcm(x,y) = [x,y] = X \/ Y
(x,y) * [x,y] = x*y = X + Y adding corresponding multiplicites

so it seems we should be able to write a descending translation table:

gcd --- intersection
lcm --- union
---precendence break
sum --- some thing
difference --- some other thing
---precedence break
product --- + (multiset sum)
quotient --- - (multiset difference)
---precedence break
exponentiation --- * (defined?)
some other thing --- /
---precedencde break
some other thing --- ^ etc.

5) Is there software like MACSYMA or Mathcad that will let you work
with both representations in the same document? Softwre that
facilitates writing "both ways at the same time?"

I'm off to Mathworld and Wikipedia to find out about this.

Doug Goncz
Replikon Research
Seven Corners, VA 22044-0394
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Robert B. Israel
science forum Guru


Joined: 24 Mar 2005
Posts: 2151

PostPosted: Sun Jul 16, 2006 9:53 pm    Post subject: Re: *unique* prime factorizations; the fundamental theorem of arithmetic Reply with quote

In article <1153043621.514503.92790@m73g2000cwd.googlegroups.com>,
Doug Goncz <DGoncz@aol.com> wrote:
Quote:
1) Has the fundamental theorem of arithmetic got a proof?

Hint: what's the definition of a theorem?

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
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porky_pig_jr@my-deja.com1
science forum Guru Wannabe


Joined: 08 May 2005
Posts: 102

PostPosted: Sun Jul 16, 2006 10:16 pm    Post subject: Re: *unique* prime factorizations; the fundamental theorem of arithmetic Reply with quote

Doug Goncz wrote:
Quote:
1) Has the fundamental theorem of arithmetic got a proof?

It's better be one ... or we would have to demote it to the
'fundamental axiom of arithmetic' :-)

Probably you can find the proof in some number-theoretical textbooks.
The proof I saw was in Apostol's Analysis. Nice and elegant proof,
using the strong induction principle.
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Proginoskes
science forum Guru


Joined: 29 Apr 2005
Posts: 2593

PostPosted: Mon Jul 17, 2006 5:15 am    Post subject: Re: *unique* prime factorizations; the fundamental theorem of arithmetic Reply with quote

porky_pig_jr@my-deja.com wrote:
Quote:
Doug Goncz wrote:
1) Has the fundamental theorem of arithmetic got a proof?

It's better be one ... or we would have to demote it to the
'fundamental axiom of arithmetic' :-)

Probably you can find the proof in some number-theoretical textbooks.
The proof I saw was in Apostol's Analysis. Nice and elegant proof,
using the strong induction principle.

Yes; it's the one I use in a course that I teach, when we do induction.
It's a very nice use of strong induction.

Uniqueness can also be proven with strong induction, once you have a
prime that divides both representations.

--- Christopher Heckman
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DGoncz@aol.com
science forum Guru Wannabe


Joined: 25 Oct 2005
Posts: 122

PostPosted: Mon Jul 17, 2006 8:13 am    Post subject: Re: *unique* prime factorizations; the fundamental theorem of arithmetic Reply with quote

Robert Israel wrote:
Quote:
In article <1153043621.514503.92790@m73g2000cwd.googlegroups.com>,
Doug Goncz <DGoncz@aol.com> wrote:
1) Has the fundamental theorem of arithmetic got a proof?

Hint: what's the definition of a theorem?

Ah. IIRC, iIt's a proposition which has been proved.

As distinct from a proposition which has been evaluated.

Like

a | b /\ a | c --> a | (b+c) as distinct from

a=2; b=4; c=6; 2|4 /\ 2|6 --> 2 | (4+6)?

Doug
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