Category Archives: number theory

Mersenne primes and the Lucas-Lehmer test

Posted on January 17, 2013 by

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 example, is … Continue reading →

Posted in arithmetic, computation, famous numbers, iteration, modular arithmetic, number theory, primes | Tagged , , , , | 3 Comments

Book review: The Irrationals

Posted on October 25, 2012 by

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 expressed as a ratio of integers ? … Continue reading →

Posted in books, number theory, review | Tagged , , , | 4 Comments

Fibonacci multiples, solution 1

Posted on June 9, 2012 by

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 twists!). Pick some arbitrary and … Continue reading →

Posted in fibonacci, modular arithmetic, number theory, pattern, pictures, proof, sequences | Tagged , , , | 5 Comments

Fibonacci multiples

Posted on May 15, 2012 by

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 things planned. But … Continue reading →

Posted in arithmetic, challenges, fibonacci, number theory, pattern | Tagged , | 12 Comments

u-tube

Posted on January 10, 2012 by

[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 numbers, More fun with infinite decadic … Continue reading →

Posted in computation, convergence, infinity, iteration, modular arithmetic, number theory, programming | Tagged , , , , | 2 Comments

Fun with repunit divisors: more solutions

Posted on November 17, 2011 by

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 repunits, every … Continue reading →

Posted in arithmetic, iteration, modular arithmetic, number theory, primes, programming, proof, solutions | Tagged

Fun with repunit divisors: proofs

Posted on November 16, 2011 by

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 some … Continue reading →

Posted in iteration, modular arithmetic, number theory, pattern, primes, proof | Tagged , , , , | 1 Comment

Fun with repunit divisors

Posted on November 11, 2011 by

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 was also a problem on … Continue reading →

Posted in arithmetic, challenges, modular arithmetic, number theory, primes | Tagged , , | 16 Comments

More fun with infinite decadic numbers

Posted on October 28, 2011 by

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 numbers). Last time I left you … Continue reading →

Posted in arithmetic, infinity, number theory | Tagged , , , , | 4 Comments

Infinite decadic numbers

Posted on October 19, 2011 by

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, for example, and are at … Continue reading →

Posted in arithmetic, convergence, infinity, number theory | Tagged , , | 7 Comments