Blog Backlog: Happy Birthday to the Big E!

Back in September Lexi put out the word to Emmett's friends and family: Dear Everyone, The Big Guy is turning 70! We're making a video to show at his party. Please send us your good wishes in .MOV format, one minute or less, and please include yourself singing "Happy Birthday!"

What followed was first a trickle, then a torrent, of endearing and entertaining footage, reflecting the deep affection and wry appreciation felt by Emmett's wide-ranging, far-flung admirers. Taken together (we realized as we knitted the clips together late on the eve of the party) this eclectic collection could only have been created to honor this one particular Emmett; there was no chance anyone would ever mistake our video for any of the other YouTube clips bearing similar titles (though it was kind of tempting to slip a couple of those adorable babies into our montage...)

We presented this to Emmett at a dinner party in his honor, hosted by the Josh and Claire (thank you!) and attended by a small but enthusiastic group of extended family members (along with some extremely adorable kittens.) Note to those of you who sent in longer videos that did not appear in their entirety: Emmett got the full versions, so he heard every word. Also not included in the video was the following, received via e-mail from Laurent Hodges:

The Integer 70

Mathematicians claim that every integer (1, 2, 3, etc.) is interesting for one or more reasons, because if one wasn’t interesting, that would necessarily make it interesting. But 70 has many interesting properties, three of which I will discuss in honor of Emmett’s 70th birthday.

First, 70 is an abundant integer, a property it shares with about one-fourth of all the integers. An abundant integer is one that is smaller than the sum of its proper divisors (its integer divisors smaller than itself), while a perfect integer like 6 equals the sum of its proper divisors (like 6 equals 1 plus 2 plus 3) and a deficient integer is greater than the sum of its proper divisors (like 10, which is greater than 1 plus 2 plus 5}. The proper divisors of 70 are 1, 2, 5, 7, 10, 14, and 35, which add to 74, which is larger than 70. In this connection I might note that every proper divisor of 70 is a deficient integer while every multiple of 70 (meaning 140, 210, 280, 350, etc.) is an abundant integer. Pretty nifty, no?

Second, you cannot remove one or more proper divisors of 70 from the sum of 74 to make 70. An abundant integer with this property is called by mathematicians a “weird integer.” In fact, 70 is the smallest weird integer, and the next largest are 836 and 4030, which are much larger.

Third, 70 has a property which makes it absolutely unique. If you take the series of integers

1, 2, 3, 4, 5, ...

and form their squares

1, 4, 9, 16, 25, ...

and then sum these squares

1, 5, 14, 30, 55, ...

it is not until you reach 24 that the sum of the squares is itself a square, 4900, which is the square of 70. In fact, 70 is the only integer which is square root of the sum of the squares of the integers from 1 up to some integer N, which in this case is 24.

There’s lots more interesting facts about the integer 70, but I don’t have room on this page to write them down.

Here's to Emmett: newly-minted septuagenarian, scholarly beach bum, musical athlete, beloved uncle, all-around geeky cool dude... Also abundant, weird, and absolutely unique. We love you!