From the BBC:
34,969 is 187 squared, but what is the significance of 187..?
Toby Lewis noted that 187 is the total number of tiles in a Scrabble game, speculating that the Count might have counted them.
David Lees noticed that 187 is the product of two primes - 11 and 17 - which makes 34,969 a very fine number indeed, being 11 squared times 17 squared. What, he asked, could be lovelier?*
And Simon Philips calculated that 187 is 94 squared minus 93 squared - and of course 187 is also 94 plus 93.** An embarrassment of riches!
But both he and Toby Lewis hinted at darkness behind the Count's carefree laughter and charming flashes of lightning: 187 is also the American police code for murder.
* VFTS points out in the comments: "Any non-prime > 1 is a product of (unique) primes - that's the fundamental theorem of arithmetic. Euclid knew that. By coincidence it happens that 187 is a product of only 2 of them."
I'd forgotten about that, but it's quite true (by definition), you can break down any non-prime number (and most numbers are non-prime) into two or more prime numbers multiplied together:
For example:
189 = 3 x 3 x 3 x 7
191, 193 = prime numbers
195 = 3 x 5 x 13
197, 199 = prime numbers
201 = 3 x 67
203 = 7 x 29
** You can say the same for abolutely any odd number:
3 = 1 + 2 and also 2^2 - 1^1
5 = 2 + 3 and also 3^2 - 2^2
...
34,969 (to take a large odd number entirely at random) = 17,484 + 17,485 and also 17,485^2 - 17,484^2
I like the other answers though.
Once I dipt into the future
14 hours ago
4 comments:
Any non-prime > 1 is a product of (unique) primes - that's the fundamental theorem of arithmetic. Euclid knew that. By coincidence it happens that 187 is a product of only 2 of them.
As you point out, a few minutes with pencil and paper will lead you to suspect that the other properties (apart from USA police) are completely general.
Looks like numerology is flourishing
VFTS, good point, I'd forgotten about that. I'll amend accordingly.
Let X be any odd number
Then
A = X/2-1/2
B = X/2 + 1/2
A and B are consecutive integers
we can now state:
X = B + A
B - A = 1
(B + A)*1 = X
(B + A)(B - A) = X
multiplying out the brackets on the left hand side
B^2 - A^2 = X
so ANY odd number is the difference between the squares of two consecutive integers
hence
187 == 94^2 - 93^2
34969 = 17485^2 - 17484^2
etc.
Dan, that's the long way round.
It's easier to imagine (or draw) a square with side length 1 to represent 1^2.
To make it a square with side length 2, = 2^2, you add an additional square on one side and two additional squares on the resulting longer side, so you've added 1+2 = 3 squares.
To make that a three-sided square 3^2, you add two additional squares on one side and three additional squares on the resulting longer side, so you've added 2+3=5 squares.
And so on.
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