Arthur Lowe would be proud of this little bit of Math....

It is called historically "Waring's Problem".

This is a little subject that has fascinated me for a while, maybe perhaps (tautology ahoy!) it took me another while to understand what the question actually was.

Basically, the point in question is about the integers (whole numbers).

We all know what a power is do we not? A "power" of a number is the result of multiplying that number by itself so many times. For instance , the number 3 "to the power of " three is 27. That is , 27 is three 3's multiplied together. 1024 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2, or, to put it in "power" language, 1024 is 2 to the power 10 (ten 2's multiplied together). We say 4 is "2 squared" and 8 is "two cubed".

Waring's problem looks at these "powers" a little more deeply.

Take the number 11, for example.

How many squares can you find to sum to 11?

Let's see.....11 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1.

Since 1 = 1 squared ( 1 = 1 x 1 ), it takes 11 squares to sum to 11.

No. it does not! I can do it in only five!

11 = (2 x 2) + (2 x 2) + 1 + 1 + 1.

HOLD ON, I can do better than that. I can do it in three....

11 = (3 x 3) + 1 + 1.

Now those are the basics. Waring proposed that all integers can be summed to AT MOST four squares.

Another example. 23.

NOW it starts to get interesting.

23 = 1 + 1 + 1 +.........(23 squares)

23 = 4 + 4 + 4 + 4 + 4 + 1 + 1 + 1 (8 squares)

23 = 9 + 9 + 1 + 1 + 1 + 1 + 1 (7 squares)

23 = 9 + 9 + 4 + 1 (4 squares)

23 CANNOT be summed by less than 4 squares. (Try it!) It is impossible.

This is the problem for squares (numbers multiplied by themselves twice). Waring was not satisfied. He generalised the problem by saying that all integers large enough (I chose 11 because it requires only three squares) need only FOUR squares to add together. In other words, 4 is the lower limit.

What about cubes? (numbers multiplied by themselves three times, such as 3 x 3 x 3 above)? What is the lower limit for them?

Turns out, it is 9. Try it with 23. It also turns out that 23 is the smallest number to require all 9 cubes. Try that as well!

Waring didn't stop there. He supposed that you need just 19 fourth powers (2 x 2 x 2 x 2, etc) is sufficient for all larger integers.

Research has gone on to prove that you need at most 37 fifth powers, all the way up to a staggering 1,051,899 20th powers.

The list goes on, and smacks of Godel yet again. That is, we can't seem to be able to prove the list, just research it, and yet it seems there is no end. Another 'Indecipherable Prepostion', to quine a phrase.

I have often been asked where do these musings come from , and my usual answer is they are like batch programs run on mainframes at night, or virus checkers run in background mode on pc's. The musings are always there, they just surface from time to time. I admit I had to look up the 20th power limit, but that is the only fact that I did look up. All the rest I have learned from casual reading, tv, etc, and mere thought.

Springer's "Main Waring" Final Thought ~~

" Don't Panic! Don't Panic! "

(Only Brits would suss this allusion)

It's a "Mys"tery

Just like many a sundry Oriental Language, Korean, or Hang-gul, sometimes "mis"ses something in the translation of simple texts.

I mean, I have seen "mis"-translations of simple directions but the above has to be an award winner, found above the stairs to the upper floor of the local fast-food chain here (Jung-ma "Lotteria").

I think it was meant to say "Be careful when carrying your meal upstairs".

Wonderful!

Apologies to my Good Korean Friend "Miss" Tori, or shall we say "Enigma"?