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Infinite decadic numbers
To recap: we’ve now defined the decadic metric on integers by where is not divisible by 10, and also . According to this metric, two numbers are close when their difference is decadically small. So,...
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More fun with infinite decadic numbers
This is the sixth in a series of posts on the decadic numbers (previous posts: A curiosity, An invitation to a funny number system, What does “close to” mean?, The decadic metric, Infinite decadic...
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Fun with repunit divisors
In honor of today’s date (11/11/11), here’s a fun little problem (and some follow-up problems) I’ve seen posed in a few places (for example, here is a very similar problem). If I recall correctly, it...
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Fun with repunit divisors: proofs
As promised, here are some solutions to the repunit puzzle posed in my previous post. (Stop reading now if you don’t want to see solutions yet!) Prove that every prime other than 2 or 5 is a divisor of...
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Fun with repunit divisors: more solutions
In Fun with repunit divisors I posed the following challenge: Prove that every prime other than 2 or 5 is a divisor of some repunit. In other words, if you make a list of the prime factorizations of...
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u-tube
[This is the eighth in a series of posts on the decadic numbers (previous posts: A curiosity, An invitation to a funny number system, What does "close to" mean?, The decadic metric, Infinite decadic...
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Fibonacci multiples
I haven’t written anything here in a while, but hope to write more regularly now that the semester is over—I have a series on combinatorial proofs to finish up, some books to review, and a few other...
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Fibonacci multiples, solution 1
In a previous post, I challenged you to prove If evenly divides , then evenly divides , where denotes the th Fibonacci number (). Here’s one fairly elementary proof (though it certainly has a few...
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Book review: The Irrationals
The Irrationals: A Story of the Numbers You Can’t Count On Julian Havil Princeton University Press sends me lots of cool books to review! Here’s one. Remember the irrational numbers, which can’t be...
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Mersenne primes and the Lucas-Lehmer test
Mersenne numbers, named after Marin Mersenne, are numbers of the form . The first few Mersenne numbers are therefore , , , , , and so on. Mersenne numbers come up all the time in computer science (for...
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Factorization diagram posters!
I’ve finally gotten around to making a nice factorization diagram poster: You can buy high-quality prints from Imagekind. (If you order soon you should have them before Christmas! =) I’m really quite...
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MaBloWriMo 15: One more fact about element orders
I almost forgot, but there is one more fact about the order of elements in a group that we will need. Suppose we have some and we happen to know that is the identity. What can we say about the order of...
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MaBloWriMo 16: Recap and outline
We have now established all the facts we will need about groups, and have incidentally just passed the halfway point of MaBloWriMo. This feels like a good time to take a step back and outline what...
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MaBloWriMo 17: X marks the spot
Recall that we are trying to prove that if is divisible by , then is prime. So let’s suppose is divisible by . We’ll prove this by contradiction, so suppose is not prime: if we can derive a...
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MaBloWriMo 18: X is not a group
Yesterday we defined along with a binary operation which works by multiplying and reducing coefficients . So, is this a group? Well, let’s check: It’s a bit tedious to prove formally, but the binary...
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MaBloWriMo 20: the group X star
So, where are we? Recall that we are assuming (in order to get a contradiction) that is not prime, and we picked a smallish divisor (“smallish” meaning ). We then defined the set as that is,...
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MaBloWriMo 21: the order of omega, part I
Now we’re going to figure out the order of in the group . Remember that we started by assuming that passed the Lucas-Lehmer test, that is, that is divisible by . Remember that we also showed for all ....
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MaBloWriMo 22: the order of omega, part II
Yesterday, from the assumption that is divisible by , we deduced the equations and which hold in the group . So what do these tell us about the order of ? Well, first of all, the second equation tells...
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MaBloWriMo 23: contradiction!
So, where are we? We assumed that is divisible by , but is not prime. We picked a divisor of and used it to define a group , and yesterday we showed that has order in . Today we’ll use this to derive a...
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MaBloWriMo 24: Bezout’s identity
A few days ago we made use of Bézout’s Identity, which states that if and have a greatest common divisor , then there exist integers and such that . For completeness, let’s prove it. Consider the set...
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