Integers etc.

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Wednesday, August 17. 2011

From a friend:

Has it struck EVERYBODY as absolutely fascinating that difference between the squares of consecutive integers equals the sum of those two integers?

Naturally, it must also have struck everybody – well, maybe not exactly everybody – that the series formed by differences of the squares of consecutive integers is, therefore, simply a series made up of consecutive odd numbers.

If neither of these has struck you, it may be because you have not been stuck, without a book, waiting in long line to get to a bathroom.

Do you think that it’s in Sloane’s? You must remember Sloane’s The On-Line Encyclopedia of Integer Sequences?

Posted by Bird Dog at 15:35 | Comments (9) | Trackbacks (0)

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Perhaps it did strike me when i was working with integers and squares thereof; if so, was a long time ago and I have forgotten.

#1 Sam L. on 2011-08-17 16:15 (Reply)

Very interesting. I wonder what you can do with that...

Another thing about numbers that is interesting is if the digits of a number add up to a number divisible by 3, that number is divisible by 3. But then everybody knew that, right! :-)

#2 mudbug on 2011-08-17 16:57 (Reply)

Two consecutive integers are X and X+1
(X+1)^2 - X^2 = X^2 + 2X + 1 -X^2 = 2X+1= (X+1) + X
Interesting but hardly fascinating.
Twice any integer is an even number and any even integer plus 1 is an odd number. So, yeah....
Whatever lets you wait in control, I suppose.

#3 MikeinAppalachia on 2011-08-17 16:58 (Reply)

I love numbers, and have studied them for years. Never saw Sloane's before, and I've never seen this rule (though it sounds familiar). So I found this really intriguing.

#4 Rick on 2011-08-17 17:22 (Reply)

It's actually a simple "solution."

For two consecutive integers a and b:

(b-a) = 1

(b-a)(b+a) = 1.(b+a)

b^2 - a^2 = (b+a)

#5 Anna on 2011-08-17 17:51 (Reply)

The OEIS has migrated to wiki form in its own .org
http://oeis.org/

#6 rhhardin (Link) on 2011-08-17 18:03 (Reply)

That reminds me of some stuff I've been listening to in one of the Great Courses series: Great Theorems, Great Thinkers. I'm not a mathematician - in fact a lot of the math went way over my head, but the professor is engaging and the stories fascinating. I highly recommend it.

#7 weaver (Link) on 2011-08-17 18:16 (Reply)

What's more interesting is that summing the cubes of the integers from one to x equals the square of the sum of the integers from one to x.

1 + 8 + 27 + ... + x^3 = (1 + 2 + 3 + ... +x)^2

for all x.

#8 John on 2011-08-17 21:24 (Reply)

What's fascinating to me is that we humans are able to internally create and manipulate abstractions that exist nowhere else in nature, yet explain and predict natural events.

#9 PacRim Jim (Link) on 2011-08-18 04:58 (Reply)

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Wednesday, August 17. 2011

Integers etc.