Why is Today “Prime Day”?

Today is July 11, which the marketing folks at Amazon* have dubbed “Prime Day.” They’ve even created a spiffy, little banner image for it:

Ooh… pretty!

The selection of 7/11 as Prime Day was no doubt deliberate, since both 7 and 11 are prime numbers, though one has to wonder why Amazon ignored the other 52 (or 53, if it’s a leap year) dates they could have chosen:

• February
  • 2/2
  • 2/3
  • 2/5
  • 2/7
  • 2/11
  • 2/13
  • 2/17
  • 2/19
  • 2/23
  • 2/29 (some years)
• March
  • 3/2
  • 3/3
  • 3/5
  • 3/7
  • 3/11
  • 3/13
  • 3/17
  • 3/19
  • 3/23
  • 3/29
  • 3/31
• May
  • 5/2
  • 5/3
  • 5/5
  • 5/7
  • 5/11
  • 5/13
  • 5/17
  • 5/19
  • 5/23
  • 5/29
  • 5/31
• July
  • 7/2
  • 7/3
  • 7/5
  • 7/7
  • 7/11
  • 7/13
  • 7/17
  • 7/19
  • 7/23
  • 7/29
  • 7/31
• November
  • 11/2
  • 11/3
  • 11/5
  • 11/7
  • 11/11
  • 11/13
  • 11/17
  • 11/19
  • 11/23
  • 11/29

One of my favorite problems is based on the numbers 7 and 11. Here’s a modified version of it, tailored to Amazon’s special day:

An online shopper placed four items in his cart. When he checked out, his credit card was charged $7.11. Shortly thereafter, a programmer realized there was an error in the code, and total price had been calculated by multiplying the prices of the four items. The customer service department was about to alert the customer to the error, but the programmer informed them that the total price would have still been $7.11 if the prices had been added. No harm, no foul.

There was no sales tax. What was the cost of each item?

Good luck! Happy shopping!

* No, Amazon did not pay me to write a blog post about Prime Day.

Math Problem for 2017

Happy New Year! Welcome to 2017.

Here are some interesting facts about the number 2017:

• It’s prime. (Okay, so that fact isn’t very interesting. But just you wait…)
• Insert a 7 between any two digits of 2017, and the result  is still a prime number. That is, 27,017, 20,717, and 20,177 are all prime. (See? Told you it was gonna get better.)
• The cube root of 2017 is approximately 12.63480759, which uses all ten digits 0‑9, and 2017 is the least positive integer that has this property. (Mind blown yet?)
• The decimal expansion of 2017²⁰¹⁷ has 6,666 digits.
• 2017 = 44² + 9²
• 2017 = 12³ + 6³ + 4³ + 2³ + 1³ = 10³ + 9³ + 6³ + 4³ + 2³

If you need some more, check out Matt Parker’s video.

Sorry, no video from me. But in honor of our newly minted prime year, I have created a problem for you to solve.

In the area model below (not to scale), the area of the five blue regions is indicated by the number inside the rectangle. What is the area of the yellow region with the question mark inside?

Sorry, I don’t give answers. Feel free to have at it in the comments.

Prime Time at NCTM Minneapolis

This afternoon, I’ll be presenting “Experience the Math Practices with Games and Online Tools” at the NCTM Regional Conference in Minneapolis. So if you unwittingly find yourself in the Minneapolis Convention Center at 1:30 p.m. today, please stop by.

But how cool is this? My session is #210, and today is Friday, November 13. Awesome, huh?

Wait, maybe you don’t see it:

210 = 2 × 3 × 5 × 7

Friday, November 13 = 11/13

Yeah, that’s right! The factors of my session number combined with today’s date are the first six prime numbers. You don’t have to be a math dork to appreciate that! (Though it doesn’t hurt.)

Why is 6 afraid of 7?

I assume it’s because 7 is a prime number, and prime numbers can be intimidating.

Thanks to Castiel from Supernatural for that new twist on an old classic.

 

Fishin’ for an April Fools Math Joke

My friend and former boss Jim Rubillo sent me the following email last night:

I am cleaning, and I found this book that you might want: One Million Random Numbers in Ascending Order. Do you want it, or should I throw it away?

Seemed like an odd book, and I thought he might have gotten the title wrong since RAND Corporation published A Million Random Digits with 100,000 Normal Deviates in 1955. (Incidentally, you should visit that book’s page on www.amazon.com, where you’ll find many fantastic reviews, such as, “Once you’ve read it from start to finish, you can go back and read it in a different order, and it will make just as much sense as your original read!” from Bob the Frog, and, “…with so many terrific random digits, it’s a shame they didn’t sort them, to make it easier to find the one you’re looking for,” from A Curious Reader.)

Knowing that Jim is a stats guy, it seemed plausible. A little confused, I wrote back:

Is it literally just a list of random numbers? If so, I’ll pass. But if there’s something more interesting about it, then maybe?

His response?

It’s a sequel to The Complete Book of Even Primes.
Gotcha!

And so it goes, with April Fools even afflicting the math jokes world.

Thank goodness he didn’t tape a fish to my back.

In Your Prime

I’ve got a prime number trick for you today.

1. Choose any prime number p > 3.
2. Square it.
3. Add 5.
4. Divide by 8.

Having no idea which prime number you chose, I can tell you this:

The remainder of your result is 6.

Pretty cool, huh?

I will now fill a bunch of space with quotes and jokes about prime numbers to prevent you from seeing the spoiler explanation below. But you can skip straight to the bottom if you’re not interested in the other stuff or if you just can’t control yourself.

Mark Haddon, author of The Curious Incident of the Dog in the Night-time, wrote the following:

Prime numbers are what is left when you have taken all the patterns away. I think prime numbers are like life. They are very logical, but you could never work out the rules, even if you spent all your time thinking about them.

(Incidentally, if you haven’t read that book, you should. Amazon reviewer Grant Cairns said it better than I could: “The integration of the mathematics into the fiction is better than any other work that I know of. The overall effect is a beautiful story that any maths fans will find hard to read without the tissue box close at hand.”)

Israeli mathematician Noga Alon said that he was interviewed on Israeli radio, and he mentioned that Euclid proved over 2,000 years ago that there are infinitely many primes. As the story goes, the host immediately interupted him and asked:

Are there still infinitely many primes?

And of course there’s this moldy oldie:

Several professionals were asked how many odd integers greater than 2 are prime. The responses were as follows:
Mathematician: 3 is prime, 5 is prime, 7 is prime, and by induction, every odd integer greater than 2 is prime.
Physicist: 3 is prime, 5 is prime, 7 is prime, 9 is experimental error, 11 is prime, …
Engineer: 3 is prime, 5 is prime, 7 is prime, 9 is prime, 11 is prime, …
Programmer: 3 is prime, 5 is prime, 7 is prime, 7 is prime, 7 is prime, …
Marketer: 3 is prime, 5 is prime, 7 is prime, 9 is a feature, …
Software Salesperson: 3 is prime, 5 is prime, 7 is prime, 9 will be prime in the next release, …
Biologist: 3 is prime, 5 is prime, 7 is prime, the results for 9 have not yet arrived…
Advertiser: 3 is prime, 5 is prime, 7 is prime, 11 is prime, …
Lawyer: 3 is prime, 5 is prime, 7 is prime, there is not enough evidence to prove that 9 is not prime, …
Accountant: 3 is prime, 5 is prime, 7 is prime, 9 is prime if you deduct 2/3 in taxes, …
Statistician: Try several randomly chosen odd numbers: 17 is prime, 23 is prime, 11 is prime, …
Professor: 3 is prime, 5 is prime, 7 is prime, and the rest are left as exercises for the student.
Psychologist: 3 is prime, 5 is prime, 7 is prime, 9 is prime but tries to suppress it, …
Card Counter: 3, 5, and 7 are all prime, but I prefer 21.

---

Explanation of the Prime Number Trick

We are trying to show that (p² + 5) mod 8 = 6. This is equivalent to showing that (p² ‑ 1) mod 8 = 0, or that (p + 1)(p ‑ 1) is divisible by 8.

Because p > 3 and is prime, then either p = 1 mod 4 or p = 3 mod 4. Consequently, it must be the case that (a) p + 1 = 2 mod 4 and p ‑ 1 = 0 mod 4 or (b) p ‑ 1 = 2 mod 4 and p + 1 = 0 mod 4. That is, both numbers will be even, and at least one of them will be a multiple of 4. For either (a) or (b), the product (p + 1)(p ‑ 1) will be a multiple of 8. Q.E.D.

Celebrity Sighting at Math Meeting

I have the pleasure of serving on the advisory committee for the Math Midway 2 Go, a traveling exhibit of the Museum of Mathematics. I get to see a lot of cool stuff.

When it opens on December 15, MoMath will be the only museum of mathematics in North America. If you happen to find yourself in Manhattan, check it out. The exhibits are really fun.

One of the exhibits in the Math Midway 2 Go is a number line with ornaments hanging from each number. For instance, a square ornament hangs from the numbers 1, 4, 9, 16, …, and a symbol that looks like an atom hangs from 2, 3, 5, 7, 11, 13, … (the atom symbol was used because “prime numbers are the building blocks of the number system”). However, I was not able to identify the symbol that hangs from the following numbers:

3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30,
32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96

I’ve been told that the symbol is a compass (the kind for drawing circles, not for orienteering). Unfortunately, that hint didn’t help me to identify the sequence of numbers. Do you know what the sequence is? **

A recent meeting of the advisory committee was held at a private school in NYC, and a number of parents were waiting in the hallway when our meeting ended. As I walked by, one of the parents stood up quickly, and I accidentally brushed against her. “Oh, I’m sorry, ma’am,” I said. She turned to look at me, and I looked back. First, I noticed how tall she was. Then I noticed something else. “Oh,” I said, “you’re Brooke Shields.” Turns out her kids go to this school. She smiled politely at my recognition.

She was dressed in casual clothes, and she was just there to pick up her kids. I didn’t want to be a nuisance, so I just said, “Have a great day.”

How cool is that? Go to a math meeting, meet a celebrity! And one I had a crush on when I was 13, no less!

** The sequence is the number of sides for constructible polygons, which are regular polygons that can be drawn with a straightedge and compass.

Prime Number Problem and Gender Bias

This morning, my friend AJ called to ask for help in solving a problem from his ten-year-old daughter’s homework. When he explained his dilemma, the first thing I did, of course, was laugh. “Wow,” I said. “You really aren’t as smart as a fifth-grader, are you?”

AJ and his daughter are both intelligent, and his daughter loves math. The problem they were trying to solve was this:

What is the units digit of the product of the first 21 prime numbers?

You can use this list of prime numbers if you need some help. As a hint, the 21st prime number is 73 (which, incidentally, is the Chuck Norris of numbers).

Once you solve the problem, of course, you realize that the problem would have the same answer if asked as follows:

What is the units digit of the product of the first n prime numbers, for n > 3?

This made me think that this could be a good problem for the classroom. Have all students randomly generate a positive integer, and then have them solve the problem above using their random number to replace n. It would be impactful for students to see that everyone gets the same answer; and those who multiplied things out might be compelled to look for a pattern and figure out why everyone got the same answer.

But then I realized: this problem is gender biased. Well, maybe. The problem asks for the units digit of the product of the first 21 prime numbers. The choice of 21 was very deliberate, I’m sure. It’s small enough that an industrious student might actually try to calculate the product. In my experience, female students are more industrious than males and therefore more likely to do the computation. But the number is large enough that male students, who are lazy like I am, will think, “That’s too much work. There’s got to be a trick!”

I mentioned to AJ that if a larger number were chosen — for instance, if it involved the product of the first 1,000 prime numbers — then it might be more obvious that students ought to look for a pattern. “You haven’t met my daughter,” he said. “She’d still try to compute it.”

You may think my assertion is crazy. There is nothing in the problem that appears inherently biased against females.

A few years ago, the AAUW published a report about gender bias in math questions. One of the selected questions was something like, “What is the value of n if n + 2 = 7?” Despite the neutrality of the content, girls scored significantly lower than boys on this question, so it was deemed to be biased. (Sorry, I wasn’t able to find a reference to the report. If anyone knows the report to which I’m referring, please share in the comments.)

Further, FairTest claims that the gender gap all but disappears on all types of questions except multiple choice when other question types were examined on Advanced Placement tests. What is it about multiple choice questions that makes them implicitly unfair to females? I have no idea.

It’s Back to Prime Time

On Saturday, I turned 41 years old. I’ve been looking forward to this for a while. It’s a prime year, and its twin prime is two years away. In between, I’ll be a number of years that is “the answer to life, the universe, and everything.”

Forty-one is also cool because f(x) = x² + x + 41 is a prime-generating function. That is, f(1) = 43, f(2) = 47, f(3) = 53, and so on.

What is the first value of x for which x² + x + 41 is not prime?

The following image might help you answer that question. The number 41 appears in the center, and consecutive positive integers then proceed in a spiral. Notice that all of the numbers highlighted in yellow are prime. A pattern of primes continues along the diagonal — at least for a little while.

It also turns out that 41 is the smallest number whose cube is the sum of three cube numbers in two different ways:

41³ = 2³ + 17³ + 40³ = 6³ + 32³ + 33³ = 68,921

And 41 is the sum of the first six prime numbers:

2 + 3 + 5 + 7 + 11 + 13 = 41

At 41, I still feel young. But you know you’re an old mathematician when…

• You report your age in hexadecimal. (I’m only 29!)
• You’re not dead, but you’ve lost most of your functions.
• The distance you walked to school as a kid is directly proportional to your age.
• Your age can be described as “countably infinite.”
• You regularly go off on tangents.
• The phrase “pulling an all nighter” means not getting up to pee.
• When asked your age, you reply, “I’m in the 99th percentile.”
• You use the term surd, and you know how to calculate its value on a slide rule.

Prime Time

One of the great joys of my current job is that I get to visit math classes. This is awesome, and I am incredibly grateful to the teachers who invite me to their classrooms. I’ve thought about returning to the classroom myself, but visiting is much better — I get to see magic and interact with kids, but I don’t have to worry about correcting misbehaviors, creating or grading tests, or filling out report cards.

I recently witnessed several great classes at Tincher Prep, a K-8 school in California. The students were the most collectively polite group of kids I’ve ever met, and the faculty was filled to capacity with intelligent, dedicated professionals. Students in first grade measured things with paper cut-outs of their foot, to get an appreciation for why we have standard measures. Kindergarten kids happily sang number songs and then counted by 5’s to figure out that it was the 75th day of the school year. Students in a middle school class were jumping out of their seats with excitement when playing a review game. In every class I visited, students were excited to be learning. What an awesome environment!

In one classroom, students were given the following assignment:

Complete this list of the first 10 prime numbers:
1, 2, ___, ___, ___, ___, ___, ___, ___, ___

John Derbyshire claims that Henri Lebesque was the last mathematician who considered 1 to be a prime number. The primary reason it should not be considered a prime number is that the Fundamental Theorem of Arithmetic — which states that every integer greater than 1 can be represented as the product of a unique set of prime numbers — will not hold. It also causes a problem with Euler’s Totient Function: for prime numbers, φ(n) = n – 1, but this rule is violated if 1 is considered a prime number.

The teacher who posed this problem to students, however, shouldn’t feel bad for including 1 as a prime number. Lots of professionals have trouble figuring out which numbers are prime…

• Mathematician: 3 is prime, 5 is prime, 7 is prime, and by induction, every odd integer greater than 2 is prime.
• Physicist: 3 is prime, 5 is prime, 7 is prime, 9 is an experimental error,
  11 is prime, …
• Engineer: 3 is prime, 5 is prime, 7 is prime, 9 is prime, 11 is prime, …
• Programmer: 3 is prime, 5 is prime, 7 is prime, 7 is prime, 7 is prime, …
• Salesperson: 3 is prime, 5 is prime, 7 is prime, 9 — we’ll do the best we can, …
• Software Salesperson: 3 is prime, 5 is prime, 7 is prime, 9 will be prime in the next release, …
• Biologist: 3 is prime, 5 is prime, 7 is prime, results have not yet arrived for 9, …
• Lawyer: 3 is prime, 5 is prime, 7 is prime, there is not enough evidence to prove that 9 is not prime, …
• Accountant: 3 is prime, 5 is prime, 7 is prime, 9 is prime if 2/3 is deducted for taxes, …
• Statistician: Let’s try several randomly chosen numbers: 17 is prime, 23 is prime, 11 is prime, …
• Professor: 3 is prime, 5 is prime, 7 is prime, and the rest are left as an exercise for the student.
• Computational Linguist: 3 is an odd prime, 5 is an odd prime, 7 is an odd prime, 9 is a very odd prime, …
• Psychologist: 3 is prime, 5 is prime, 7 is prime, 9 is prime but tries to
  suppress it, …
• Casino Card Counters: 3 is prime, 5 is prime, and 7 is prime, but I’ll take 21 over any of them.

Deep Math Thoughts

I’ve been thinking about a lot of things lately…

Instead of having “answers” on a math test, they should just call them “impressions,” and if you get a different impression, so what? Can’t we all just get along?

If you think that dogs can’t count — let him watch you put two biscuits in your pocket, and then only give him one.

If we stop teaching students about numbers less than zero, do you think there would be a positive impact on education?

A prime rib cannot be cut with a steak knife, because it is only divisible by itself and one.

A math professor is someone who talks in other people’s sleep.

A mathematician is someone who will begin a sentence with, “As everyone knows,” and then finish it with something he just learned.

And a famous quotation…

Never trust any quote you find on the Internet.
– Abraham Lincoln, Second Inaugural Address