*unique* prime factorizations; the fundamental theorem of arithmetic

DGoncz@aol.com · science forum Guru Wannabe Joined: 25 Oct 2005 Posts: 122

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

DGoncz@aol.com · science forum Guru Wannabe Joined: 25 Oct 2005 Posts: 122

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:

Robert B. Israel · science forum Guru Joined: 24 Mar 2005 Posts: 2151

In article <1153043621.514503.92790@m73g2000cwd.googlegroups.com>,
Doug Goncz <DGoncz@aol.com> wrote:

porky_pig_jr@my-deja.com1 · science forum Guru Wannabe Joined: 08 May 2005 Posts: 102

Doug Goncz wrote:

Proginoskes · science forum Guru Joined: 29 Apr 2005 Posts: 2593

porky_pig_jr@my-deja.com wrote:

DGoncz@aol.com · science forum Guru Wannabe Joined: 25 Oct 2005 Posts: 122

Robert Israel wrote:

Google