Category Archives: primes

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

Triangunit divisors and quadratic reciprocity

Posted on March 20, 2011 by

Recall that the triangunit numbers are defined as the numbers you get by appending the digit 1 to the end of triangular numbers. Put another way, where denotes the th triangular number, and the th triangunit number. The challenge, posed … Continue reading →

Posted in arithmetic, modular arithmetic, number theory, primes, proof | Tagged , , , , | 5 Comments

Book Review: The Mystery of the Prime Numbers

Posted on November 7, 2010 by

Several months ago, Matthew Watkins sent me a review copy of his new book, Secrets of Creation Volume One: The Mystery of the Prime Numbers. It’s taken me a while to get around to reviewing it, but not for lack … Continue reading →

Posted in books, number theory, primes, review | Tagged , , , | 3 Comments

MMM #33: Super divisible

Posted on May 25, 2009 by

This week’s Monday Math Madness is up at Wild About Math!. Looks like a fairly accessible problem this week: What’s the prime factorization of the smallest whole number that is divisible by all integers from 1 up to and including … Continue reading →

Posted in challenges, links, number theory, primes | Tagged , ,

More on repetend lengths

Posted on January 27, 2009 by

In a previous post, I noted that the length of the repetend (repeating portion of the decimal expansion) of a fraction with prime denominator p is at most p-1, and in fact divides p-1. I also said: In fact, there’s … Continue reading →

Posted in group theory, number theory, pattern, primes | Tagged , , , , | 6 Comments

Two Very Large primes

Posted on September 17, 2008 by

As promised, I can now reveal the identity of the two newly discovered Mersenne primes. The smaller of the two, discovered on September 6 by Hans-Michael Elvenich in Langenfeld, Germany, is an 11,185,272-digit number which you can download here. The … Continue reading →

Posted in computation, famous numbers, links, primes | Tagged , , , | 3 Comments

New Mersenne primes!

Posted on September 15, 2008 by

The Great Internet Mersenne Prime Search just announced not one, but two new Mersenne primes! You might also recall the last time they announced a new prime, in September 2006, so these new primes were found almost exactly two years … Continue reading →

Posted in computation, famous numbers, links, primes | Tagged , , , , | 1 Comment

Perfect numbers, part III

Posted on November 27, 2007 by

This is the last in a series of posts about perfect numbers. A quick recap of the series so far: in part I, I defined perfect numbers as positive integers n for which , where denotes the sum of the … Continue reading →

Posted in algebra, number theory, open problems, primes, solutions | 11 Comments