"Anti-closure (+) of integer p-th powers, prime p>2" --- (sci.math 7jan96) ---

 Fermat's Anti-Closure {n^p} (+) as Generator of the Naturals  (sci.math 28nov98)

All those FLT (Fermat) postings may have "snowed under" some rather
obvious observation:

If the sum of two p-th powers is NOT a p-th power, then clearly the
p-th powers (prime p>2) are a good set of generators of the naturals
under addition ;-)  --- Consider powersums: \sum (n_i)^p  (i=1..k) ---

"Waring" (1770):
   The sum of how many g(p) p-th powers reach all naturals under (+) ?
   For p=2 this is known to be g(2)=4 (Fermat, Euler, Lagrange).

For Waring mod p^k ("Waring for Residues"), have a look at:

          http://www.iae.nl/users/benschop/nwb1.dvi
intro's:  http://www.iae.nl/users/benschop/fewago.htm
          http://www.iae.nl/users/benschop/func.htm  
   (on the role of function composition in arithmetic analysis)

to see that all pairsums {x^p+y^p} mod p^k cover half the units group
in Z(.) mod p^k, for sufficient precision k ('critical precision' K_p)
depending on p (K_p =2 for most primes, but e.g. K_11 =3, K_73 =4 ).

And eventually: each residue mod p^k (any prime p>2, any k>1) is the
sum of at most four p-th power residues. The corresponding 'core' A_k
(k \geq K_p) subgroup of units mod p^k of order |A_k|=p-1, an
extension of Fermat's Small Thm subgroup A_1 mod p, has the additive
property that all pairsums a+b mod p^k are distinct (a,b \in A_k),
apart from commutation.  - Let me know what you think of it.

--
Ciao, Nico Benschop.             | AHA: One is Always Halfway Anyway
http://www.iae.nl/users/benschop | xxxxxxxxxxxxxxx1.1xxxxxxxxxxxxxxx
http://www.iae.nl/users/benschop/inertia.htm

(-- FLT inequality -------- Look at it another (positive) way ------;-)

Subject:  Re: what is the logic behind zero     (sci.math 16aug02)
-------
Teacherjh wrote:
>
> [...]
> Here's another mystery for you.
>
> 2+2 is the same as 2x2 which is the same as 2^2.      ..[*]
>   However, this is not true of any other number.      ..[*]
> Zero comes close, but 0^0 will start a thread of discussion here,
> whereas 2^2 will not.  So, what's the magic of 2?     -- Jose

Re[*]: No 'magic', really.

NB:  By definition (notational agreement):

      a + a = 2 x a,  and:  a x a = a^2

    Replace 'a' by '2' and you get the 'magic'...

The clue is that you combine just two numbers into one number:
a 'binary' operation (having two operands), the essence of the
main concept of 'closure' in math.

The complementary concept is 'generation' which is in extreme form:
anti-closure, as in x^p + y^p =/= z^p, the sum of two equal types
(p-th powers, p>2) is NOT of that type. If *that* holds, you have
an efficient set of generators (here: under addition), which is
rather exceptional and not easy to find.

-- NB -  http://home.iae.nl/users/benschop/anti-cl.htm