Repunit prime

In the United States, today is abbreviated 10/31. Define the nth repunit number as

R_n = \frac{10^n-1}{9} = 1111\dots1,

a base-10 number consisting of n consecutive 1s. For example,

R_1 = 1

R_2 = 11

R_3 = 111

R_4 = 1,111,

and so on.

It turns out that R_{1031} is the largest known prime repunit number.

Source: http://mathworld.wolfram.com/Repunit.html

Pizza Hut Pi Day Challenge: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on the 2016 Pizza Hut Pi Day Challenge.

Part 1: Statement of the problem.

Part 2: Using the divisibility rules for 1, 5, 9, 10 to reduce the number of possibilities from 3,628,800 to 40,320.

Part 3: Using the divisibility rule for 2 to reduce the number of possibilities to 576.

Part 4: Using the divisibility rule for 3 to reduce the number of possibilities to 192.

Part 5: Using the divisibility rule for 4 to reduce the number of possibilities to 96.

Part 6: Using the divisibility rule for 8 to reduce the number of possibilities to 24.

Part 7: Reusing the divisibility rule for 3 to reduce the number of possibilities to 10.

Part 8: Dividing by 7 to find the answer.

 

Euler and 1,000,009

Here’s a tale of one the great mathematicians of all time that I heard for the first time this year: the great mathematician published a mistake… which, when it occurs today, is highly professionally embarrassing to modern mathematicians. From Mathematics in Ancient Greece:

In a paper published in the year 1774, [Leonhard] Euler listed [1,000,009] as prime. In a subsequent paper Euler corrected his error and gave the prime factors of the integer, adding that one time he had been under the impression that the integer in question admitted of the unique partition

1,000,009 = 1000^2 + 3^2

but that he had since discovered a second partition, namely

1,000,009 = 235^2 + 972^2,

which revealed the composite character of the number.

See Wikipedia and/or Mathworld for the details of how this allowed Euler to factor 1,000,009.

Factoring Mersenne “primes”

I love hearing and telling tales of legendary mathematicians. Today’s tale comes from Frank Nelson Cole and definitely comes from the era before calculators or computers. From Wikipedia:

On October 31, 1903, Cole famously made a presentation to a meeting of the American Mathematical Society where he identified the factors of the Mersenne number 267 − 1, or M67. Édouard Lucas had demonstrated in 1876 that M67 must have factors (i.e., is not prime), but he was unable to determine what those factors were. During Cole’s so-called “lecture”, he approached the chalkboard and in complete silence proceeded to calculate the value of M67, with the result being 147,573,952,589,676,412,927. Cole then moved to the other side of the board and wrote 193,707,721 × 761,838,257,287, and worked through the tedious calculations by hand. Upon completing the multiplication and demonstrating that the result equaled M67, Cole returned to his seat, not having uttered a word during the hour-long presentation. His audience greeted the presentation with a standing ovation. Cole later admitted that finding the factors had taken “three years of Sundays.”

My Favorite One-Liners: Part 92

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

This is one of my favorite quote from Alice in Wonderland that I’ll use whenever discussing the difference between the ring axioms (integers are closed under addition, subtraction, and multiplication, but not division) and the field axioms (closed under division except for division by zero):

‘I only took the regular course [in school,’ said the Mock Turtle.]

‘What was that?’ inquired Alice.

‘Reeling and Writhing, of course, to begin with,’ the Mock Turtle replied; ‘and then the different branches of Arithmetic — Ambition, Distraction, Uglification, and Derision.’

My Favorite One-Liners: Part 18

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them. This is a quip that I’ll use when a theoretical calculation can be easily confirmed with a calculator.

Sometimes I teach my students how people converted decimal expansions into fractions before there was a button on a calculator to do this for them. For example, to convert  x = 0.\overline{432} = 0.432432432\dots into a fraction, the first step (from the Bag of Tricks) is to multiply by 1000: How do we change this into a decimal? Let’s call this number x.

1000x = 432.432432\dots

x = 0.432432\dots

Notice that the decimal parts of both x and 1000x are the same. Subtracting, the decimal parts cancel, leaving

999x = 432

or

x = \displaystyle \frac{432}{999} = \displaystyle \frac{16}{37}

In my experience, most students — even senior math majors who have taken a few theorem-proof classes and hence are no dummies — are a little stunned when they see this procedure for the first time.

To make this more real and believable to them, I then tell them my one-liner: “I can see that no one believes me. OK, let’s try something that you will believe. Pop out your calculators. Then punch in 16 divided by 37.”

Indeed, my experience many students really do need this technological confirmation to be psychologically sure that it really did work. Then I’ll tease them that, by pulling out their calculators, I’m trying to speak my students’ language.

TI1637

See also my fuller post on this topic as well as the index for the entire series.

 

My Favorite One-Liners: Part 14

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them. This quip is similar to the “bag of tricks” one-liner, and I’ll use this one if the “bag of tricks” line is starting to get a little dry.

Sometimes in math, there’s a step in a derivation that, to the novice, appears to make absolutely no sense. For example, to find the antiderivative of \sec x, the first step is far from obvious:

\displaystyle \int \sec x \, dx = \displaystyle \int \sec x \frac{\sec x + \tan x}{\sec x + \tan x} \, dx

While that’s certainly correct, it’s from from obvious to a student that this such a “simplification” is actually helpful.

To give a simpler example, to convert

x = 0.\overline{432} = 0.432432432\dots

into a decimal, the first step is to multiply x by 1000:

1000x = 432.432432\dots

Students often give skeptical, quizzical, and/or frustrated looks about this non-intuitive next step… they’re thinking, “How did you know to do that?” To lighten the mood, I’ll explain with a big smile that I’m clairvoyant… when I got my Ph.D., I walked across the stage, got my diploma, someone waved a magic wand at me, and poof! I became clairvoyant.

Clairvoyance is wonderful; I highly recommend it.

The joke, of course, is that the only reason that I multiplied by 1000 is that someone figured out that multiplying by 1000 at this juncture would actually be helpful. Subtracting x from 1000x, the decimal parts cancel, leaving

999x = 432

or

x = \displaystyle \frac{432}{999} = \displaystyle \frac{16}{37}.

In my experience, most students — even senior math majors who have taken a few theorem-proof classes and hence are no dummies — are a little stunned when they see this procedure for the first time. I learned this procedure when I was very young; however, in modern times, this procedure appears to be a dying art. I’m guessing that this algorithm is a dying art because of the ease and convenience of modern calculators. As always, I hold my students blameless for the things that they were simply not taught at a younger age, and part of my job is repairing these odd holes in their mathematical backgrounds so that they’ll have their best chance at becoming excellent high school math teachers.

For further reading, here’s my series on rational numbers and decimal expansions.

15-square puzzle

From the category “This Is Completely Useless”: here’s what a 15-square puzzle looks like when you arrange the tiles in order of how many factors they have.

puzzle15

My Mathematical Magic Show: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The links below show the mathematical magic show that I’ll perform from time to time.

Part 1: Introduction.

Part 2a, Part 2b, and Part 2c: The 1089 trick.

Part 3a, Part 3b, and Part 3c: A geometric magic trick.

Part 4a: Part 4b, Part 4c, and Part 4d: A trick using binary numbers.

Part 5a, Part 5b, Part 5c, and Part 5d: A trick using the rule for checking if a number is a multiple of 9.

Part 7: The Fitch-Cheney card trick, which is perhaps the slickest mathematical card trick ever devised.

Part 8a, Part 8b, and Part 8c: A trick using Pascal’s triangle.

Part 9: Mentally computing n given n^5 if 10 \le n \le 99.

Part 6: The Grand Finale.

And, for the sake of completeness, here’s a recent picture of me just before I performed an abbreviated version of this show for UNT’s Preview Day for high school students thinking about enrolling at my university.

magician

 

Engaging students: Reducing fractions to lowest terms

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Madison duPont. Her topic, from Pre-Algebra: reducing fractions to lowest terms.

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How can this topic be used in your students’ future courses in mathematics or science?

Reducing fractions to lowest terms can be applied to future mathematics topics such as ratios and proportions, and scientific topics such as chemistry or physics. Ratios can be represented as fractions and are not typically reduced to lowest terms because they represent relationships of two subjects using numbers. Being able to reduce these ratios can help students better identify the underlying relationship and apply this relationship to other aspects of the math problem, such as problems using unit price or map scales. Proportions relate to the concept of reducing fractions to lowest terms when using cross-multiplication. Having both sides of the proportion reduced to lowest terms makes the cross-multiplication much easier to compute and derive a final reduced answer. Chemistry uses fractions reduced to lowest terms with topics, like stoichiometry, that use potentially small and large numbers in several ratios that are multiplied together to obtain a final converted and reduced answer. Physics often uses ratio-like formulas and problems that are applied to real-world scenarios, which typically require fractions reduced to lowest terms because answers like miles per one hour are the goal. All of these topics use concepts of reducing fractions to lowest terms to more easily accomplish problems using a series of fractional computations, or to get an answer that is in terms of a single unit or most reduced so that it makes sense to real-world application.

 

 

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How does this topic extend what your students should have learned in previous courses?

This topic extends previously learned topics such as concepts of unique prime factorizations, greatest common divisor, manipulating fractions, and multiplication facts. The concept of unique prime factorizations greatly aids students in finding the greatest common divisor, which is used to find the greatest factor of the value of both the numerator and denominator. Next, manipulation of fractions is used to properly divide the numerator and denominator by the greatest common divisor. This process of dividing both parts of the fraction utilizes multiplication facts as well to determine what the answer to the division problem on both the top and bottom of the fraction would be. These previously learned concepts are all subtle and important applications when reducing fractions to lowest terms.

 

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How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

 

 

This video reminded me of many students that I have tutored or encountered in classrooms that were determined that a calculator was all they needed when doing math. Applied to reducing fractions to lowest terms, this video is extremely relevant in displaying that technology cannot be the only source of intelligence when thinking mathematically. Reducing fractions with extremely large numbers or numbers that do not have well-known factors can seem exhausting or impossible. Punching several factors of the numerator and denominator into a calculator attempting to reduce numbers with each common factor, and then not being sure of whether the fraction appearing on their screen is truly in the most reduced form surely indicates the technology is not the only way of solving the problem. Many students hop on a procedural escalator when beginning varying types of problems (in addition to reducing fractions to lowest terms) using memorized steps, punching calculator buttons, feeling comfortable, until suddenly—there is a horribly unattractive fraction halting their progress. This is when using mathematical problem solving skills such as reducing the numerator and denominator by the greatest common divisor or checking to see that the numerator and denominator are relatively prime becomes pertinent. Using these conceptual skills can save someone that is stuck waiting for a calculator to do the work for them, or that has given up on finishing a problem because it seems impossible or difficult, from thinking they are incapable of working out a problem efficiently and successfully. This video highlights the importance of being capable of knowing when it is time to take the effort to climb the stairs to reach your destination.

 

References:

“Stuck on an Escalator” Video link:

https://www.youtube.com/watch?v=VrSUe_m19FY

found via Google video search