Following are some activities relating to number theory.
The items marked with are the contributions of the Summer 2000 participants.

 Fermat's Enigma and Pythagorean Triples The Golden Ratio Perfect Numbers Square Numbers A Pascal Triangle Activity The Sieve of Eratosthenes A Magic Square Activity Fermat's Enigma and Pythagorean Triples

Pythagoras' Theorem states that given a right triangle with legs a and b and hypotenuse c,

a2  +   b2  =  c2. Numbers that fit this pattern are called Pythagorean triples. The chart below shows several of these Pythagorean triples. Study the pattern in the chart to find the next two sets of triples.

 a b c 3 4 5 6 8 10 9 12 15

Will any of the triples satisfy Fermat's Last Theorem - a3  +  b3  =  c3?

What about a4  +  b4  =  c4?

The following formulas can be used to generate numbers that will satisfy Pythagoras' Theorem. Be sure that x > y and that one of them is odd and the other is even.

a = x2 - y2     b = 2xy     c = x2 + y2

Here is an example: Let x=7 and y=6.

 a = x2-y2 b = 2xy c = x2 + y2 a=72-62 b=2*7*6 c=72 + 62 a = 49-36 b = 2*7*6 c = 49 + 36 a = 13 b = 84 c = 85
Notice that one leg and the hypotenuse (longest side) are consecutive integers. When you try some numbers on your own, see if your answers always include consecutive integers.

Tip: To get numbers a, b, and c which have no factors in common, make sure your numbers x & y have no common factors, for example 5 & 6 or 5 & 8, but not 5 & 10.

Contributed by Charlene Evans

Reference:

Collins, William, et al. Mathematics: Applications and Connections, Course 3. Glencoe/McGraw Hill. Ohio. 1998

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The Golden Ratio

Activities to Discover the Golden Ratio in Nature

Remember: If you divide the larger number by the smaller number the Golden Ratio should be close to 1.618 and if you divide the smaller number by the larger number the Golden Ratio should be close to 0.618.

1. Have students pair off and make the following measurements on each other using a tape measure and record their data:
1. Measure from the top of the head to the bottom of the neck and measure from the bottom of the neck to the belly button.
2. Measure from the belly button to the bottom of the knee cap and from the bottom of the knee cap to the floor.
3. Measure from the top of the head to the belly button and from the belly button to the floor.
4. Now have the students divide the larger number of each set of data by the smaller number or the other way around and see what they come up with.

2. Have each student bend their index finger to form a U shape. Now measure from the 1st knuckle to the 2nd knuckle and then from the 2nd knuckle to the 3rd knuckle.

Now have the students divide the data just as they did in the first activity.

3. This activity would work well around Easter. You need hard boiled eggs unless you want to have a real mess on your hands. Measure the length and height of different eggs.

Now divide like you have in the other activities.

If you want more Golden Ratio measurement in nature try these just using the face and head of your students.

1. Measure the height and width of your head.
2. Measure from the top of the eyes to the bottom of your nose and from the bottom of the nose to the bottom of you chin.
3. Measure from the top of the eyes to the bottom of your nose and from the bottom of the nose to the lips.

Divide these and see how close you come to the Golden Ratio.

Contributed by James Means

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Perfect Numbers

Ask students to find the first three perfect numbers. (Students could also set up a computer program to check their answers.)

They will need to find a number n of the form (2n-1)*2n-1 with the factor 2n-1 being prime.

The first three perfect numbers are 6, 28, and 496.

Contributed by Kristen Shelton

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Square Numbers

Square Numbers Activity

Grade Level- Middle School

Purpose- Use at beginning of a square number lesson to test student's knowledge and show concrete example of square number.

Objective- The learner will be able to

1. define a square number,
2. construct the dimensions of a square number, and
3. determine if a number is a square number.

Resources- graph paper; colored pencil

Activity- Write the numbers 1- 20 on the board. Explain to the students that they are trying to determine which of these numbers are square numbers. For #1, have them shade in 1 box. Is this a square number? ( 1 x 1) For #2, shade in 2 boxes. Square number? Students will continue working on graph paper in groups of three.

Tell students that if they determine a pattern, write it down. Constantly monitor groups to make sure they understand. Square numbers will have the pattern 1 x 1, 2 x 2, 3 x 3……… The square numbers are 1, 4, 9, 16…….

Contributed by Andrea Reynolds

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A Pascal Triangle Activity

Objectives
The students will recognize patterns.
The students will add large digit numbers.
The students will learn the history of Pascal's Triangle.

Time 45minutes

Materials
copies of Pascal's Triangle (2 for each student)
transparency of Pascal's Triangle
crayons

Focus  Tell a small story of Pascal and how he used the triangle.

Guided Practice
Pass out Pascal's Triangle to each student and have it displayed on the overhead. If there is no overhead, make a large Pascal's Triangle on a large piece of butcher paper.

The students color the left and right edge of the triangle.

Together in class, explain how to add two units using color. If you have one blank circle and one colored circle, you color in the circle below the joining circles.  If you have two of the same circles beside each other, the circle below remains blank.
Do the triangle together in class at least for 10 rows and then the students can finish on their own. Go over the design of the triangle. Never skip a row, you must do each row. Independent Practice
Give the students a new sheet of Pascal's Triangle and tell the students to put numbers instead of colors. Start with ones on the sides.

Reflection
What are some things you have seen with Pascal's Triangle in the design?

Contributed by Tina Gonzales

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The Sieve of Eratosthenes

The Sieve of Erathosthenes is the most effective way to find all of the small prime numbers. Your first step is to make a list of all the integers less than or equal to n (n being the largest number you want to check) and also greater than one. Next, you need to cross out all of the multiples of the prime numbers less than or equal to the square root of n. Finally, the numbers you have left are the prime numbers.

For example, let’s let n be equal to 26.

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Now, in our list 2 is the first prime. Therefore, cross out all the multiples of two. You are left with:

3 5 7 9 11 13 15 17 19 21 23 25

Now, the first number (3) left is our first odd prime. Therefore, cross out all the multiples of three. You are left with:

5 7 11 13 17 19 23 25

Now, the first number left (5) is the second odd prime. Therefore, cross out all the multiples of five. You are left with:

7 11 13 17 19 23

Now, the first number left (7) is the third odd prime. Therefore, cross out all the multiples of seven. You are left with:

11 13 17 19 23

Now, the first number left (11) is the fourth odd prime. Therefore, cross out all the multiples of eleven. You are left with:

13 17 19 23

Now, the first number left (13) is the fifth odd prime. Therefore, cross out all the multiples of thirteen. You are left with:

17 19 23

Now, the first number left (17) is the sixth odd prime. Therefore, cross out all the multiples of seventeen. You are left with:

19 23

Now, the first number left (19) is the seventh odd prime. Therefore, cross out all the multiples of nineteen. You are left with:

23

Finally, the first number left (23) is the eighth odd prime. Therefore, cross out all the multiples of twenty-three. You are left with:

Nothing

Now, we know that there are eight odd primes less than twenty-six. There is also an even prime (2) less than twenty-six. That makes a total of 9 primes less than twenty-six.

Click on this link to read about Eratosthenes.

Here is a pseudo-code that can be used to generate all prime numbers.

```Eratosthenes(n)  {
a :=0
for i :=2 to n do a[i] :=1
p:=2
while p^2<=n do {
j:=2p
while (j<=n) do {
a[j]:=0
j:=j+p
}
repeat p :=p+1 until a[p]=1
}
return (a)
}
```

Contributed by Jeremy Troutman

References:

URL:http://www.utm.edu/research/primes/glossary/SieveOfEratosthenes.html 6-13-00 8:00 PM.

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Making Magic Squares
On Changing Squares into Doughnuts

Unraveling the mystery of the magic square and learning how to construct them is not very difficult. The truths follow...

It is beneficial to understand what a magic square is. According to a definition by Allen Adler

“A magic square is an arrangement of the numbers from 1 to n2 in an nxn matrix, with each number occurring only once, such that the sums of any row, any column, or any main diagonal is the same.”
The sum will satisfy n(n2 + 1)/2.

Read on to find out how to make you own...
The simplest magic square is the 1x1 magic square, with the only entry being the number 1.

There aren’t any 2x2 magic square because the above equation cannot be satisfied with the numbers 1 through 4.

Each odd numbered magic square, 3x3, 5x5, etc., are constructed in a similar manner. When there are odd number of columns and rows, start with the number 1 in the middle column, first row. From here, continue filling the square by this method: move up one square and to the right one square.

Three things could cause you to feel as though your magic went astray. There are ways to overcome the puzzles that ail you:

1. You are at the top row of your square and have no place to go.
*Pretend the top edge is pasted to the bottom edge and come up through the bottom, still moving up and to the right. 2. You are at the right side of the square and, again, have no place to go.
*Pretend the right edge is pasted to the left edge. Continuing to move up and to the right, the four would be in the left most column, third row. Fill in the number five as well. 3. Now, look what you have done, where will you put the 6? For this situation, when there is a number already occupying the space, you must abandon the rules, but don’t give up! Place the six directly under the five. Remember this rule for the future. Through hard work and concentration, you will eventually have worked enough magic to make a square like this one: You may have some trouble placing number 16. Follow each of the above rules and you will succeed!

How is making magic squares like making doughnuts?

Refer to rules 1 and 2 that were given above that helped you do your magic. Those rules will help you again. When you reached the edge of the square you were instructed to pretend to paste the top edge to the bottom edge, matching up the columns. If you were to do this, you would make a cylinder.

You also pasted the left side to the right side, matching up each row to itself. If you do this after constructing the cylinder, the ends of the cylinder would be pasted together.

Now say: You are truly a magician. You changed the magic square into a doughnut!!

Please refer to http://mathforum.com/alejandre/magic.square/adler for more information on magic squares.

Contributed by Cynthia Schmidt

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