Thursday, October 13, 2011

In Your Prime

"There's something strange about that boy. All he wants to talk about is chemistry and numbers."

Barbara, my morning companion at the Bagel Shop in Little Silver has shifted the monologue, err, conversation, from her recent bout with shingles to a weekend visit by her ten-year-old grandson.

"The whole weekend he's pointing out what things are made of...
Grandma, did you know that Grandpa's shirt is made of polyethylene terephthalate. It's a thermoplastic polymer that can synthesized by the esterification reaction between terephthalic acid and ethylene glycol. Blah, blah, blah...
What are you supposed to do with a boy like that?"

Barbara looks at me waiting for a reaction.

"You said he talks about numbers?"

"Oh, don't get me started! Every time someone mentions a number, he starts spouting off facts about it. Yesterday, a friend wished my daughter a happy thirty-ninth birthday. Before she could even respond, Daniel starts spouting facts about the number thirty-nine...
You know, many people think that thirty-nine is a prime number. It sounds like a prime number but it's not. First of all, the sum of its digits is divisible by three, which of course means that it's divisible by three: three and thirteen, both of which are prime. Blah, blah, blah...
I tell you there's something strange about the boy."

I'm thinking, "He's right. Thirty-nine does sound like a prime number though it's not."

Barbara, continues talking and I start thinking about numbers...

Prime numbers are the ones that are only divisible by themselves and one. Non-prime numbers can be divided by other, smaller numbers, some of which are prime and some of which are not. Fifty-one is another number that sounds like it's prime, but it's not. It's divisible by three and seventeen. The three part is easy. Any time the sum of the digits is divisible by three (e.g., five plus one equals six), the number itself is divisible by three.

It's easy to eliminate lots of candidates for prime number status. First, there are all the even numbers which are divisible by two. That's like half of them right away. The ones that are divisible by five are easy to spot. They all end in five. That leaves you numbers that end in one, three, seven or nine.

Then there's the little trick with adding up the digits which eliminates a third of the candidates. Twenty-nine adds up to eleven, thirty-nine to twelve, and forty-nine to thirteen. Only thirty-nine is divisible by three. Only twenty-nine is prime (forty-nine is the square of seven).

Yeah, thinking about numbers is fun. I particularly like multiples of eleven and thirteen. The patterns are fun to watch: 11, 22, 33, 44, 55, 66, 77, 88... or 13, 26, 39, 52, 65, 78, 91... Aren't they cool? It's like finding a groove that can grind on forever.

The patterns in number sequences are similar to harmonic and rhythmic sequences in music. Of course with harmony you have multiple multiplicative series stacked on top of one another. Any note in a chord has a relationship with each of the other notes and although the relationship is reciprocal, it's not interchangeable; a flatted ninth is fine on top, but invert it and it sounds pretty bad. And then theres the... Oops. There I go. Blah, blah, blah...

Happy Thursday,


  1. Oh yes, numbers and patterns are most definitely fun. Reminds me of 1729, the Hardy-Ramanujan number? When the Indian mathematician Ramanujan was in hospital during his ill-fated stay in England, his friend the English mathematician Hardy paid him a visit. To make small talk, he told Ramanujan that the taxi that brought him was numbered 1729, which he thought was a dull number and hoped wasn't unlucky, since it is divisible by 13. To which Ramanujan replied, "No, Hardy, 1729 is actually a very interesting number! It is the smallest number expressed as the sum of two cubes in two different ways".

    And now, looking up Wikipedia, I find 1729 has another interesting property:
    it is one of only four positive integers (with the others being 81, 1458, and the trivial case 1) which, when its digits are added together, produces a sum which, when multiplied by its reversal, yields the original number:

    1 + 7 + 2 + 9 = 19
    19 × 91 = 1729


  2. By the way,
    1729 = 1^3 + 12^3 and also 9^3 + 10^3


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