Monthly Archives: February 2013

Prime Patterns

There is no known formula for finding prime numbers. All numbers must be tested to determine whether they are prime or not. There are some interesting patterns that pop up in prime numbers, but you never know how long they will hold.

For example:

PrimePatterns1

PrimePatterns2

These types of numbers, where all the digits are the same number except one, are known as near repdigit primes.

Another pattern can be found when adding and subtracting factorials (!). The factorial of a number is the product of that number and all other positive integers less than that number. So for example, 4! = 4 x 3 x 2 x 1.

Alternating adding and subtracting factorials, as shown below, yields primes numbers until you get to 9!.

PrimePatterns3

If you want to see if a number is prime, or find out the factors of number that isn’t prime (those number are called composite), you can check numbers from 1 to 100,000,000 at cpnumbers.com.

Twin Primes

 In an earlier post, I explained what prime numbers are and how to find them using a sieve. In this post, I will tell you a little about twin primes.

Twin primes are pairs of prime numbers separated by a non-prime (composite) number. So, using  our sieve (shown below with prime numbers highlighted, and twin primes in red with darker yellow highlighting), we can see that the twin primes up to 100 are:

  • 3, 5
  • 5, 7
  • 11, 13
  • 17, 19
  • 29, 31
  • 41, 43
  • 59, 61
  • 71, 73

TwinPrimeSieve

The twin prime conjecture states that there are an infinite number of these pairs. This proposition is called a conjecture because it has not been proven. This video from Nova, says that the twin prime conjecture was proposed by Euclid, but it seems that some believe it was actually first proposed by Alphonse de Polignac.

February 21st is Introduce a Girl to Engineering Day

Introduce a Girl to Engineering Day is devoted to teaching girls about the field of engineering, a field traditionally dominated by men. This event aims to address one of the causes of this gender disparity, a lack of familiarity with the field.

For more information, resources, and hands-on activities for K-12 kids, go the Introduce a Girl to Engineering Day webpage.

Introduce a Girl to Engineering Day, 2013, is dedicated to the memory of Dr. Sally Ride.  Dr. Ride was America‘s first woman in space and founder of Sally Ride Science, a company devoted to promoting K-12 science education.

189px-Ride-s

Star Polygons

Drawing stars is a fun activity. Once you learn how, you can make some really cool looking stars with any number of points you want.

Also, drawing stars can be related to a number of mathematical concepts, such as:

  • multiples and factoring
  • prime numbers
  • symmetry
  • geometry terms (polygon, vertex, edge)

Vi Hart has a great video about drawing stars. She goes pretty quickly, so I’ve included instructions below.

A star polygon is a figure formed by connecting every qth point out of p regularly spaced points lying on a circle.  The star polygon is denoted by {p,q}. So for example, the 5-pointed star, that we are all familiar with, is a {5,2} star, because it has 5 points (p), and you make it by connecting every 2nd (qth) point.

Web

To create the star polygon:

  1. Start with a circle.
  2. Decide how many points you want your star to have, and place that number of equally spaced dots around that circle.
  3. Choose a starting point.
  4. Draw a line to the point q spaces over.
  5. Repeat until you are back to a starting point.
  6. If you reach a starting point and not all points have been covered, lift the pencil, skip over 1 point, and repeat the process. As shown for the {6,2} star below:

Web

I have a worksheet, with these instructions and pre-printed circles and points, available here.

So, how do these cool stars relate to all those other math concepts?

  • Stars where you can touch all points without lifting your pencil, such as the {5,2} star, are called regular polygon stars. In these cases, p and q are relatively prime, which means they have no common factors. Another example is the {10,3} star. There are no numbers (besides 1) that divide evenly into both 10 and 3.
200px-Star_polygon_10-3

{10,3}

  • In cases that create “asterick” stars (such as {8,4}) or regular polygon stars (the ones where you had to lift your pencil), q is a divisor of p. Or you can say the p is a multiple of q. The {9,3} star consists of 3 regular triangles.
StarPolygon6_3

{8,4}

200px-Star_polygon_9-3

{9,3}

  • The number of lines of symmetry for a star polygon is equal to the number of vertices (points).
76026_5_pt_star_mth

5-pointed star with lines of symmetry

Prime Numbers: What are they and how do you find them?

A prime number is a natural number (i.e., the counting numbers: 1, 2, 3, 4…), greater than 1, that has no divisors except 1 and itself. For example, 5 is prime because no numbers divide into it evenly except 1 and 5. The number 4 is not prime, because in addition to 1 and 4, it can be divided evenly by 2. Numbers that are not prime are called composite numbers.

The most basic method of checking whether or not a number is prime (a property called primality) is called trial division. Basically, this consists of dividing the number (we’ll call it n) by every integer (or whole number) that is greater than 1 and less than or equal to the square root of the number. If any of these divisions results in an integer, then n is not a prime. So, using a small number as an example, to test if 13 is a prime number, we would divide it by all integers from 1 to 3 (the square root of 13 is around 3.6). None of these divisions result in an integer, so 13 is prime.

Another method for finding all primes up to a given number, is called a prime number sieve. The oldest example is the sieve of Eratosthenes (shown in the animated gif below). This sieve is useful for relatively small primes. To make your own sieve of Eratosthenes up to 100, start with a number grid up to 100 (10 row, with 10 numbers in each row row).

  1. Cross out 1, since prime numbers must be greater than 1.
  2. The first prime number is 2.  Cross out all multiples of 2 (the even numbers).
  3. The next prime number is 3. Cross out all multiples of 3 (in other words, count by three’s: 3, 6, 9, 12…).
  4. The next prime is 5 (4 has already been crossed out). Cross out all multiples of 5.
  5. The next number remaining is 7 (the next prime). Cross out all remaining multiples of 7.
  6. All remaining numbers are prime numbers. (No other numbers have multiples below 100 that have not already been crossed out.)

You can download my worksheet with these instructions and a grid here.

Sieve_of_Eratosthenes_animation

You may have read something about prime numbers in the news recently. The largest known prime, was discovered by a mathematician at University of Central Missouri, Curtis Cooper. This newly discovered prime number is 17,425,170 digits long! Of course, finding primes this large require powerful computers and algorithms more advanced than the sieve above, but I’ll leave that for a future post.