Wow, anonimnystefy!

Hi;

Hi noelevans,

Hi phrontister!

That's using your noodle and your computer!  Turns out to be a great combination for lots of problem
solving.  I bet Gauss, Newton, Euler, etc. would have given a lot to have the computational power
that we have today.  They might not have left us any math to discover!

One of the characteristics of primes larger than two is that there is only one way to write them as
a difference of squares.  Composites always have at least two ways to do it.  Each factorization of
a number greater than two corresponds to a difference of squares factorization.  For example
15=1x15=8^2-7^2  and 15=3x5=4^2-1^2.  21=1x21=11^2-10^2  and 21=3x7=5^2-2^2.
So a number like 27 which has 2 factorizations has 2 ways to write it as a difference of squares.

Primes and perfect squares are neat, multifaceted numbers.

Have a blessed evening!

Edit:  Oops!  Limit the above discussion to odd non-square composite numbers.  Even numbers and
         odd perfect squares may have fewer difference of square factorizations, especially if we do
         not allow zero as one of the numbers.  Example 4=4x1=(5/2 + 3/2)(5/2 - 3/2) but these are
         not integral.  Also 4=2x2=(2+0)(2-0) won't do if we disallow zero.
         

         For 81=9x9 we have no difference of square factorization if we disallow zero.
         But 81=1x81=(41+40)(41-40) and 81=3x27=(15+12)(15-12).

         for 9=1x9=(5+4)(5-4)  but 9=3x3=(3+0)(3-0) won't work if zero is disallowed.
         So the square of a prime has only one difference of squares factorization IF we
         disallow zero.

Last edited by noelevans (2012-12-08 14:59:47)

Zeckendorf representations --- VERY INTERESTING!  Thanks for pointing them out.

Actually I was just thinking of a base as a single integer >1.  I've never seen a SET being a base.
Wiki points out that this property is characteristic of the Fibonacci numbers so no other set of
numbers has this property ascribed to the Fibonacci numbers.

My question is a bit tricky.  Any positive integral base greater than 1 could satisfy the conditions I
mentioned.  The coefficients could be negative integers (introduced by a many named Colson in
                                                                                    - --
1729) for example  481 = 1x1000 - 5x100 - 1x10 -9x1 = 1519 where the bars above the digits
mean subtract instead of adding.  So 481 is written without any coefficients >1.

But if we require that the absolute value of the integral coefficients not be greater than 1 then
base 2 and one other base will satisfy this.

I had this message written earlier and I previewed it but forgot to submit it, so I lost it.

Live and learn! 

Got it, bobbym!

Hi;

Okay!
Two official answers are:
121
484
Bobbym, Phrontister, you are correct!

bobbym wrote:

484

phrontister wrote:

121

Hi;

Bobbym, Phrontister, you are correct!

I like phrontister's answer better. Where that other gets his answer from I can only speculate.

Maybe, but he probably would miss the board!

I meant thump them in with a hammer...not throw them.

Hi;

I meant thump them in with a hammer...not throw them.

That would be more accurate.