Posts tagged ‘prime’
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.
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.
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?
It’s a sequel to The Complete Book of Even Primes.
And so it goes, with April Fools even afflicting the math jokes world.
Thank goodness he didn’t tape a fish to my back.
I’ve got a prime number trick for you today.
- Choose any prime number p > 3.
- Square it.
- Add 5.
- 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.
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.
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?
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.
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.