Category Archives: modular arithmetic

Post without words #26

Posted on May 6, 2019 by Brent

Posted in modular arithmetic, number theory, posts without words | Tagged Euler, grid, totient | 2 Comments

Chinese Remainder Theorem proof

Posted on April 10, 2019 by Brent

In my previous post I stated the Chinese Remainder Theorem, which says that if and are relatively prime, then the function is a bijection between the set and the set of pairs (remember that the notation means the set ). … Continue reading →

Posted in modular arithmetic, number theory | Tagged Chinese, proof, remainder, theorem | 2 Comments

More words about PWW #25: The Chinese Remainder Theorem

Posted on April 5, 2019 by Brent

In a previous post I made images like this: And then in the next post I explained how I made the images: starting in the upper left corner of a grid, put consecutive numbers along a diagonal line, wrapping around … Continue reading →

Posted in modular arithmetic, number theory, posts without words | Tagged Chinese, grid, remainder, theorem, torus | 3 Comments

A few words about PWW #25

Posted on March 9, 2019 by Brent

In my previous post I made images like this: What’s going on? Well, first, it’s easy to notice that each grid starts with in the upper-left square; is one square down and to the right of , then is one … Continue reading →

Posted in modular arithmetic, number theory, posts without words | Tagged Chinese, grid, remainder, theorem, torus | 4 Comments

Post without words #25

Posted on March 5, 2019 by Brent

Posted in modular arithmetic, number theory, posts without words | Tagged grid | 4 Comments

Finding the repetend length of a decimal expansion

Posted on February 8, 2019 by Brent

We’re still trying to find the prefix length and repetend length of the decimal expansion of a fraction , that is, the length of the part before it starts repeating, and the length of the repeating part. In my previous … Continue reading →

Posted in computation, group theory, modular arithmetic, number theory, pattern | Tagged decimal, expansion, group theory, rational, repeating, repetend, totient | Comments Off

MaBloWriMo 24: Bezout’s identity

Posted on November 25, 2015 by Brent

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 of all linear … Continue reading →

Posted in algebra, arithmetic, modular arithmetic, number theory | Tagged Bezout, combination, divisor, gcd, identity, linear, MaBloWriMo, proof | 2 Comments

MaBloWriMo 23: contradiction!

Posted on November 24, 2015 by Brent

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 … Continue reading →

Posted in algebra, group theory, modular arithmetic, number theory, proof | Tagged contradiction, groups, lehmer, lucas, MaBloWriMo, Mersenne, omega, order, prime, proof, test, X | 5 Comments

MaBloWriMo 22: the order of omega, part II

Posted on November 23, 2015 by Brent

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 us … Continue reading →

Posted in algebra, group theory, modular arithmetic, number theory, proof | Tagged groups, lehmer, lucas, MaBloWriMo, Mersenne, omega, order, prime, proof, test, X | 1 Comment

MaBloWriMo 21: the order of omega, part I

Posted on November 22, 2015 by Brent

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 . In … Continue reading →

Posted in algebra, group theory, modular arithmetic, number theory, proof | Tagged groups, lehmer, lucas, MaBloWriMo, Mersenne, omega, order, prime, proof, test, X | 2 Comments