Tag Archives: number theory

September 19, 2011, 8:00 am

Math Monday: What is casting out nines?

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Last week in this post, I asked for requests for math topics you’d like to read about. One person wrote in and asked:

Why don’t you enlighten us about the name “Casting Out Nines?” I learned a system in grade school with the same name –it was a way of checking multiplication and long division answers. Long before calculators.

A review please?

OK then. Casting out nines is an old-fashioned method of checking for errors in basic arithmetic problems (addition and subtraction too, not just multiplication and division). Here’s how it works, using addition as an example.

Let’s suppose I’m trying to add 32189 to 87011. I get a sum of 119200. But did I make a mistake? Do the following to check:

  1. Take the first number, 32189, and remove — “cast out” — any 9′s…

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March 1, 2009, 4:40 pm

Keeping things in context

Part of Article 131 in the first edition (1801...
Image via Wikipedia

I’ve started reading through Stewart and Tall’s book on algebraic number theory, partly to give myself some fodder for learning Sage and partly because it’s an area of math I’d like to explore. I’m discovering a lot about algebra in the process that I should have known already. For example, I didn’t know until reading this book that the Gaussian integers were invented to study quadratic reciprocity. For me, the Gaussian integers were always just this abstract construction that  Gauss invented evidently for his own amusement (which maybe isn’t too far off from the truth) and which exists primarily so that I would have something to do in abstract algebra class. Here are the Gaussian integers! Now, go and find which ones are units, whether this is a principal ideal domain, and so on. Isn’t this fun?

Well, yes, actually it is fun for me, but that’s because I like a…

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November 18, 2008, 8:50 pm

Undergraduates make a "prime" discovery

This is cool:

Westfield State College senior mathematics majors Jeffrey P. Vanasse and Michael E. Guenette, working under the direction of Mathematics Department faculty members Marcus Jaiclin and Julian F. Fleron, have made a significant new discovery in the mathematical field of number theory. They have discovered the first known example of a 3 by 3 by 3 generalized arithmetic progression (GAP).

Most easily thought of as a 3 by 3 by 3 cube (similar to a Rubik’s cube puzzle) made up of 27 primes, their discovery begins with 929 as its smallest prime ends with 27917 as its largest prime. The intervening 25 primes are constructed by adding combinations of the numbers 2904, 3150, and 7440 in an appropriately structured method.

“Such an object was known to exist and its approximate size had been loosely estimated,” Fleron said. “However, a blind search would require checking more …

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September 11, 2008, 8:13 pm

It's a prime! And another prime!

The number believed to be the 45th Mersenne prime has turned out actually to be a prime, according to GIMPS. The verification was completed on 6 September and announced on 7 September.

But in a fairly extraordinary turn of events, yet another number was submitted to the GIMPS servers as the next possible Mersenne prime on 6 September — and the initial verification shows that it is prime too! So we now have the 45th and 46th Mersenne primes discovered within two weeks of each other, which is incredible.

No word yet on the details of these primes. We’ll soon see who wins the Mersenne prime digit-guessing challenge. You can still play along with your own spreadsheet too!

August 28, 2008, 1:57 pm

Estimating the digits in a Mersenne prime — for dummies

At the end of this post, I made a totally naive guess that the recently discovered candidate to be the \(M_{45}\), the 45th Mersenne prime, would have 10.5 million digits. There was absolutely no systematic basis for that guess, but I did suggest having an office pool for the number of digits, so what I lack in mathematical sophistication is made up for by my instinct for good nerd party games. On the other hand, Isabel at God Plays Dice predicted 14.5 million digits based on a number theoretic argument. Since I am merely a wannabe number theorist, I can’t compete with that sort of thing. But I can make up a mean Excel spreadsheet, so I figured I’d do a little data plotting and see what happened.

If you make a plot of the number of digits in \(M_n\), the nth Mersenne prime, going all the way back to antiquity, here’s what you get:

The horizontal axis is n and the vertical…

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August 26, 2008, 8:05 pm

New Mersenne prime discovered?

GIMPS is reporting that on 23 August a new Mersenne prime was reported to their server. Verification began today and should take about two weeks to complete. No word on what the prime was, how many digits, etc.

The last Mersenne prime discovered was \(2^{32,582,657}-1\), back in 2006 (blogged about here) and weighed in at a whopping 9,808,358 digits. Any bets on how big this new one is, if it’s really a prime? I’m guessing 10.5 million digits. Sounds like a good occasion for a nerd office pool.

Update: Isabel at God Plays Dice likes 14.5 million digits instead, and she’s actually using math and stuff to make that estimate instead of just shooting totally in the dark like I am.

May 1, 2008, 8:03 pm

(Super-)Powers of 2

Driving in to work this morning, I suddenly felt my vision go blurry to the point where I literally couldn’t see anything. Fortunately, I was able to pull off the road into the parking lot of a small office building before causing an accident. After I stopped and waited for the blurriness to subside, the first thing I saw was the mailbox for the office building, which had a street number of: 2048. Rather than wonder what the crap was wrong with my eyesight, or frantically try to decide whether to go see a doctor on the spot, instead the first thing I thought was hey: That’s \(2^{11}\). 

Then, after making it to work with no more blurry vision attacks, I walked up to my office — the same office I have been entering and exiting since summer 2001 — and looked at the office number and saw it: room 128. Of course, I’ve never had a problem remembering my office number But for the first…

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