In a triangle with a right angle, there are 2 legs and the hypotenuse of a triangle.


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1 PROBLEM STATEMENT In a triangle with a right angle, there are legs and the hypotenuse of a triangle. The hypotenuse of a triangle is the side of a right triangle that is opposite the 90 angle. The legs of a triangle are the other shorter, perpendicular sides. The Pythagorean Theorem can be used to figure out the legs and the hypotenuse of a right triangle. The Pythagorean Theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. a + b = c leg (b) hypotenuse (c) leg (a) Figure 1. A right triangle with the legs and hypotenuse labeled. A triple is a set of 3 numbers. The average student knows usually 3 of these Pythagorean triples. They usually know the 3,4,5 triple, the 5,1,13 triple, and the 8,15,17 triple. This research deals with the exploration of Pythagorean triples. How difficult it is to get these Pythagorean Triples? In Table 1, there is a list of numbers that the calculator chose randomly. In Table, there is a list of selected. The numbers listed under a, are the lengths of the legs, the numbers listed under b are the length of the legs, and the letter c represents the length of the hypotenuse. Amanda Magli Page 1
2 Table 1. Selected Numbers Used to Find Pythagorean Triples a b c Does It Form a Triple? NO NO 1 4 NO NO NO NO NO NO NO 3 13 NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO Amanda Magli Page
3 Table. The Numbers the Calculator Randomly Picked to Find Pythagorean Triples a b c Does This Form a Ttriple? NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO YES NO NO NO NO NO NO NO NO NO NO Amanda Magli Page 3
4 Table 1 shows us that we found no numbers that formed a Pythagorean Triple. From Table, one set of numbers formed a Pythagorean triple. Based on this information, we can conjecture that it is not easy to pick out Pythagorean triples randomly or by guess and check. This paper explores Pythagorean triples and to derive a formula to get triples more efficiently. The questions that we will investigate during this report are: Is there a formula to figure out different Pythagorean Triples? How many Pythagorean triples are there? These two questions will be answered throughout this paper. RELATED AND ORIGINAL RESEARCH When the Smallest Leg is Odd The tables in this section show the steps of deriving a formula to generate Pythagorean triples using several given triples. Each table shows us, step by step, the process and hints of how to get a Pythagorean triple when the smallest leg is odd. Amanda Magli Page 4
5 Table. 3 The Perimeter of 10 Selected Right Triangles Pythagorean Triple Perimeter 3,4, =1 5,1, = 30 7,4, = 56 9,40, = 90 11,60, = 13 13,84, = 18 15,11, = 40 17, 144, = ,180, = 380 1,0, = n?,? On column 1 of Table 3, several Pythagorean triples are presented. Many patterns are formed with these Pythagorean Triplets. In the second column, the sums of the elements are shown. Each integer of the Pythagorean triple is added together to get a sum of the 3 numbers. Table 4. Alternative Method for Finding the Perimeter of Right Triangles Pythagorean Triple Perimeter Alternative Method for Finding the Perimeter 3,4, = ,1, = ,4, = ,40, = ,60, = ,84, = ,11, = , 144, = ,180, = ,0, = n?,? n(n+1) Amanda Magli Page 5
6 The third column of Table 4 shows an alternative method for finding the perimeter. The smallest element is involved as a factor in this formula for the perimeter. For example, the first Pythagorean triple is 3, 4, 5. We take the smallest integer (the 3) and multiply that by 4. Then the next largest triple we take is the smallest integer in the triple and multiply that by the next consecutive integer. So, for 5, 1, 13 we multiply 5 by 6 to get the perimeter. Now, since we know this, we can substitute n. Since we know that to find the sum of the elements we multiply the smallest integer by itself plus 1. We conjecture that the formula for finding the alternative sum of the elements is: n(n+1) Now after looking at the alternative method for finding the perimeter we are going to look at the method for finding the sum of the two larger elements. Amanda Magli Page 6
7 Pythagorean Triple Table 5. The Method for Finding the Sum of the Two Larger Perimeter Alternative The Smallest Method for Number Finding the Squared Perimeter Method for Finding the Sum of the Two Larger 3,4, = = 9 5,1, = = 5 7,4, = = 49 9,40, = = 81 11,60, = = 11 13,84, = = ,11, = = 5 17, 144, = = 89 19,180, = = 361 1,0, = = n?,? n(n+1) n n(n+1) n = n The 4 th column of Table 5 shows the smallest integer of each triple squared. As we can see it relates to the 5 th column which it demonstrates a method for finding the sum of the two larger elements. In this case, let s take the perimeter and subtract the smallest integer in the triple to get the sum of the two larger integers. For example, in the 7,4,5 triple we take = 56 Then we subtract = 49 Notice that the smallest element squared is equal to the sum of the two larger elements. Therefore we get the equation. Amanda Magli Page 7
8 n(n+1)n = n After finding the sum of the two larger elements we will now look at what the even element will be. Table 6. Finding the Even Element of the Pythagorean Triples Pythagorean Triple 3,4, = = Perimeter Alternative Method for Finding the Perimeter The Smallest Number Squared Method for Finding the Sum of the Two Larger Even Element = 5,1, = = = 1 7,4, = = = 4 9,40, = = = 40 11,60, = = = 60 13,84, = = = 84 15,11, = 5 1 = 40 5 = 11 17, 144, = 306 = 89 = ,180, = 380 = 361 = 180 1,0, _ = 46 = 441 = n?,? n(n+1) n n(n+1) n 1 n = n Amanda Magli Page 8
9 The 6 th column of Table 6 shows how to get the even element. We take the sum of the two larger elements, subtract 1, and then divide by. We add or subtract in this case to identify the remaining element. For example, we take 9, (the answer when we subtracted the sum of the elements by the smallest integer) and we subtract 1. Let s then divide it by to get 4, the even element in this triple. Also by knowing this we can figure out a formula. The n formula is: n 1 As you can see, this is how we got the even element. Now we are going to try to find the greatest element (hypotenuse). Amanda Magli Page 9
10 Table 7. Finding the Greatest Element (Hypotenuse) Pythagorean Triple Perimeter Alternative Method for Finding the Perimeter The Smallest Number Squared Method for Finding the Sum of the Two Larger 3,4, = = Even Element Greatest Element (Hypotenuse) 5,1, = 30 7,4, = 56 9,40, = 90 11,60, = 13 13,84, = 18 15,11, = 40 17, 144, = ,180, = = = = = 1 = = = = = = = = = 60 = = = 84 = = = 11 = = = 144 = = = 180 = 181 1,0, = = = 0 = n?,? n(n+1) n n(n+1) n = n 1 n + 1 n The last column in Table 7 is the column that shows the greatest element, which is the hypotenuse. We take the sum of the two larger elements, add 1 and divide by to get the Amanda Magli Page 10
11 greatest element. For example we take 5 which is the sum of the two larger elements of the 5,1,13 triple. Add 1 to 5: Then divide that by = 13 The formula that can be used to find the greatest integer in a Pythagorean triple is: n + 1 Now we can find out the formula for the legs in the 1 st column. If n is the smallest integer then the even integer is n The greatest integer (the hypotenuse) is 1 n + 1 We conjecture that these formulas generate a Pythagorean triple. Now, we can look at the last line of the table. It shows how we got that n(n+1) n = n. This was found by taking the smallest integer, n, and then multiplying that by (n +1). This gave us the sum of the 3 integers in Pythagorean Triples. We then subtracted the smallest integer from that product to get n. This was the method for finding the sum of Amanda Magli Page 11
12 the two larger elements. By using this information we found out that n 1 is the formula for deriving the even element and n + 1 is the formula for deriving the greatest element (the hypotenuse). After getting those two pieces of information, we can fill in the last row of the first column. The smallest integer is represented by n, the even n 1 n + 1 element is represented by, and the greatest element is represented by. We are going to check that these 3 formulas work before completing the final row. First we are going to take the 3 formulas and put them into the Pythagorean Theorem. x + y = z Let x = n Let y = n 1 and let z = n + 1 Substitute: n 1 n + 1 ( n) + =? n 1 n + 1 We now are going to multiply and Amanda Magli Page 1
13 n + = n 1 n 1 n + 1 n + 1? This equals n = n n 1 n n 1? Then make the expression on the left one fraction 4 4 4n + n n + 1 n + n + 1 =? 4 4 Combine like terms: n n + 1 n + n + 1 = 4 4 The expressions n, n 1, n + 1 actually do generate triples. Our conjecture is true. Now we can fill in the last row of the table. Table 8. The Last Column of Each Table n, n 1 n + 1, n 1 n + n n(n+1) n n(n+1) n = n n 1 n + 1 If you look at this column, we might think to ourselves how do we know that the answer we get won t be a fraction. We know that the hypotenuse and the even element won t be a fraction because when we subtract or add 1 to an odd number, we will always get an Amanda Magli Page 13
14 even number as the result. Even numbers are always divisible by so therefore we will never get a fraction. Here is an example using this formula: Let n = 15 The even element is The hypotenuse is 15 1 = = 113 We now know that a Pythagorean triple is 15, 11, and 113. Another example is if n = 3. The even element is The hypotenuse is 3 1 = = 65 This Pythagorean triple is therefore, 3, 64 and 65. Now since we have looked at the smallest element being odd, now we are going to look at the smallest element being even. Amanda Magli Page 14
15 When the Smallest Element is Even The next series of tables shows that if the smallest integer in a Pythagorean triple is even than a Pythagorean triple can also be formed. Table 9. Pythagorean Triples and the Associated Perimeter n Associated Triple Perimeter 3 6,8, ,15, ,4, , 35, ,48, ,63, ,80, n,?,? n,?,? In Table 8 there are the triples and the perimeters of the triples. We took the smallest even numbered element of these sets of triples and designated it n. The n is derived by taking the n and dividing it by. For example, in the 6, 8, 10 triple, 6 is the smallest even numbered element. When n = 3, 6 is that smallest even numbered element of that Pythagorean triple. n = n The perimeter was found by adding all of the numbers of each triple. An example is, = 4 Notice that the first numbers of successive triples differ by. After deriving the perimeter we will figure out a formula to form the perimeter in a different way. Amanda Magli Page 15
16 Table 10. An Alternate Formula for Finding the Perimeter of the Triangle n Associated Triple Perimeter An Alternative Way of Generating the Perimeter 3 6,8, ,15, ,4, , 35, , 48, , 63, ,80, n n,?,?. n(n+1) Table 9 shows the value of n, the associated triple, the perimeter of the triple, and an alternate formula for the perimeter of the formula without adding up the 3 numbers together. The formula that is used to express these values is n(n+1). An example of this is the triple 6,8,10. n = 3 This is because the number 6 in this triple is equal to n. Therefore the formula to get the perimeter of the triple is (6)(3 + 1) After finding an alternate formula for find the perimeter of a triangle, we are now going to look at the difference between the perimeter and the smallest side. Amanda Magli Page 16
17 Table 1. Difference Between the Perimeter and the Smallest Side n n Associated Triple Perimeter An Alternative Way of Generating the Perimeter Difference Between Perimeter and Smallest Side ,8, = ,15, = ,4, = , 35, = , 48, = , 63, = ,80, = n n n,?,?. n(n+1) n In Table 11, the 6 th column shows the smallest element subtracted from the perimeter of each triangle. Perimeter Smallest element = Sum of larger elements. In the 6,8,10 triple we take the perimeter = 4 We subtract the smallest element from it 46 =18 We can conjecture that the sum of the larger elements is n. For example, in the 8,15,17 triple, it is 4 multiplied by equals 3 which is the difference between the Amanda Magli Page 17
18 perimeter and the smallest side. Now we are going to look at the formula for the middle element. Table 1. The Formula for Generating the Middle Element n n Associated Triple Perimeter An Alternative Way of Generating the Perimeter Difference Between Perimeter and Smallest Side Formula for the Middle Element ,8, = = ,15, = 3 3 = ,4, = = , 35, = 7 7 = , 48, = = , 63, = = ,80, = = n n n,?,?. n(n+1) n? In column 7 of table 1, we find the formula for the middle element. We take n This is the same as saying n  1 For example, for the 8, 15, 17 triple we take Amanda Magli Page 18
19 3 = 15 Now we are going to show how to get the largest element. Table 13. The Formula for Generating the Largest Element n n Associated Triple Perimeter An Alternative Way of Generating the Perimeter Difference Between Perimeter and Smallest Side ,8, = ,15, = ,4, = , 35, = , 48, = , 63, = ,80, = 16 Formula for the Middle Element Formula for the Largest Element = = = 15 = = = = = = = = = = = n n n,?,?. n(n+1) n n  1 n + 1 Amanda Magli Page 19
20 In column 8 of Table 13, we find the formula for the largest element. We take the difference between the perimeter and the smallest side and add. We then divide two to get the largest element. For example, in the 10, 4, 6 triple we take 50 + = 6 We know that n + = n + 1 We conjecture that n + 1 is the formula for the largest element. Now we are going to check to see if these 3 formulas work by substituting them into the Pythagorean Theorem. x + y = z Let x = n Let y = n 1 Let z = n + 1 Substitute: (n) + (n 1) = (n + 1)? We are now going to multiply (n), (n 1) and (n + 1) (n)(n) + (n 1)(n 1) = (n + 1)(n + 1)? This equals Amanda Magli Page 0
21 4n + n 4 n n + 1 = n 4 + n + n + 1? Combine like terms n 4 + n + 1 = n 4 + n + 1 Our conjecture is true. These 3 expressions do generate Pythagorean triples. The Euclidean Method We now are going to find a third formula for finding several Pythagorean triples. We first are going to set up a column for the sum of the elements like we have done in the previous tables. Table 14. Sum of of Selected Triples Triple Sum of 4, 3, 5 1 8, 15, , 5, , 7, , 9, , 11, , 65, , 119, , 161, ?,?,? In Table 14 columns 1 and, we have a list of triples in which the even element is first. In column there is the sum of the elements. For example, in the 8, 15,17 triple we add each number together to get the sum of the elements. Now we are going to look at the sum of the odd elements. Amanda Magli Page 1
22 Table 15. Sum of the Odd Triple Sum of Sum of Odd 4, 3, , 15, , 5, , 7, , 9, , 11, , 65, , 119, , 161, ??,? Sum of Odd Expressed as Twice a Perfect Square In Table 15, column 3 we have the sum of the odd elements. The first (even) element is subtracted from the sum of the triple to generate the sum of the odd elements. For example in the 4,3,5 triple we add the two odd elements, 3 and 5, to get the sum of the odd elements. In this case the sum of the odd integers is 8. We can also relate column 3 with column 4. Notice that the sum of the odd integers is twice a perfect square. For example in the 40, 9, 41 triple the sum of the odd elements is 50. If you divide that by, the answer you will get will be a perfect square, in this case 5. Now we are going to look at the difference between the two odd elements. Amanda Magli Page
23 Triple Table 16. Difference Between the Two Odd Sum of Sum of Odd Sum of Odd Expressed as Twice a Perfect Square 4, 3, , 15, , 5, , 7, , 9, , 11, , 65, , 119, , 161, ??,? Difference Between the Two Odd In Table 16 column 5 we have the difference between odd elements. We took the odd elements from each triple and subtracted them. For example, in the 1, 5, 13 triple, we took 135 = 8 Column 4 also relates to column 5. Twice a perfect square comes up with a number in the difference between the two odd elements column. For example, in the 4, 7, 5 triple, the difference between the two odd elements is 18. If you divide 18 by you get 9 which is a perfect square. The next column that we are going to look at is the column that explains the average value of the odd elements. Amanda Magli Page 3
24 Triple Sum of Table 17. Average Value of the Odd Sum of Odd Sum of Odd Expressed as Twice a Perfect Square Difference Between the Two Odd Average Value of the Two Odd Expressed as a Perfect Square 4, 3, = 8, 15, = 4 1, 5, = 3 4, 7, = 4 40, 9, = 5 60, 11, = 6 7, 65, = 9 10, 119, = , 161, = ??,? u In column 6 table 17, we have the average value of the odd elements. For example, if we look at the 1, 5, 13 triple we add the two odd numbers in that triple = 18 Then we divide by to get the average 18 = 9 The number 9 is the same as 3. Nine is a perfect square. We can represent the average value of the odd elements with u. Now we are going to look at half of the difference of the odd elements. Amanda Magli Page 4
25 Triple Table 18. Half of the Difference of the Odd Sum of Sum of Odd Sum of Odd Expressed as Twice a Perfect Square Difference Between the Two Odd Average Value of the Two Odd Expressed as a Perfect Square Half of the Difference of the Odd Expressed as a Perfect Square 4, 3, = 1 = 1 8, 15, = 4 1 = 1 1, 5, = 3 4 = 4, 7, = 4 9 = 3 40, 9, = 5 16 = 4 60, 11, = 6 5 = 5 7, 65, = 9 16 = 4 10, 119, = 1 5 = , 161, = = ??,? u v In column 7 table 18, we explore half of the difference of the odd elements. For example, in the 4, 7, 5 triple the difference of the odd elements is 18. Therefore half of this difference gives you 9. 9 = 3 Nine is a perfect square. We can represent the half of the difference of the odd elements with v. It is now time to look at the hypotenuse. Amanda Magli Page 5
26 Triple Sum of Table 19. The Formula for Generating the Hypotenuse Sum of Odd Sum of Odd Expressed as Twice a Perfect Square Difference Between the Two Odd Average Value of the Two Odd Expressed as a Perfect Square Half of the Difference of the Odd Expressed as a Perfect Square Hypotenuse 4, 3, = 1 = , 15, = 4 1 = , 5, = 3 4 = , 7, = 4 9 = , 9, = 5 16 = , = 6 5 = , 61 7, = 9 16 = , 97 10, 119, = 1 5 = , 161, = = ??,? u v u + v Column 8 of Table 19 shows the hypotenuse of a triangle. As we can see the hypotenuse is the sum of average value of the odd elements and half of the difference of the odd elements. For example, in the 40, 9, 41 triple the average value of the odd elements is 5 and half of the difference of the odd elements is = 41 (the hypotenuse) Amanda Magli Page 6
27 Since we know this, we can label the hypotenuse as u + v since u represents the average value of the odd elements and v represents half of the difference of the odd elements. The odd leg is the next side of the triangle we are going to look at. Table 0. The Formula for Generating the Odd Leg Triple Sum of Sum of Odd Sum of Odd Expressed as Twice a Perfect Square Difference Between the Two Odd Average Value of the Two Odd Expressed as a Perfect Square Half of the Difference of the Odd Expressed as a Perfect Square Hypotenuse Odd Leg 4, 3, = 1 = , 15, = 4 1 = , 5, = 3 4 = , 7, = 4 9 = , 9, = 5 16 = , 11, = 6 5 = , 65, = 9 16 = , 119, = 1 5 = , 161, = = ??,? u v u + v u  v In Column 9 of Table 0, look at the odd leg of each triangle. The odd leg of each triangle is half of the difference of the odd elements subtracted from the average value of Amanda Magli Page 7
28 the odd elements. For example, in the 8, 15, 17 triple the average value of the odd elements is 4 and half of the difference of the odd elements is 1. Now we subtract them. 4 1 = 15 (odd leg) We can label the odd leg as u v since it is half of the difference of the odd elements subtracted from the average value of the sum of the odd elements. We are now going to look at the even leg. Amanda Magli Page 8
29 Triple Sum of Table 1. The Formula for Generating the Even Leg Sum of Odd Sum of Odd Expressed as Twice a Perfect Square Difference Between the Two Odd Average Value of the Two Odd Expressed as a Perfect Square Half of the Difference of the Odd Expressed as a Perfect Square Hypotenuse 4, 3, = 1 = ()(1) 8, 15, = 4 1 = (4)(1) 1, 5, = 3 4 = (3)() 4, 7, = 4 9 = (4)(3) 40, 9, = 5 16 = (5)(4) 60, 11, = 6 5 = (6)(5) 7, 65, = 9 16 = (9)(4) 10, 119, = 1 5 = (1)(5) 40, 161, 89 Odd Leg Even Leg = = (15)(8) uv, u v u + v u  v uv u +/v Look at the 10 th column of Table 1. Notice that the even leg is twice u times v. For example, for the 4, 7, 5 triple the hypotenuse is The 4 is represented by u and the 3 is represented by the v. If we leave out the squared for the hypotenuse we get the numbers 4 and 3. The number 4 represents u and the number 3 represents v. The even leg is therefore, twice the product of u v. In this case (4)(3) = 4 The number 4 is the even leg in this triple. Amanda Magli Page 9
30 We have now discovered the Euclidean Method. We are going to show that this works in the Pythagorean Theorem. Let x = u v Let y = uv Let z = u + v Substitute (u v ) + (uv) = (u + v )? We are now going to multiply (u v ), (uv), and (u + v ) (u v )(u v ) + (uv)(uv) = (u + v )(u + v )? This equals u 4  u v + v 4 + 4u v = u 4 + u v + v 4? Combine like terms u 4 +u v +v 4 = u 4 +u v +v 4 Our conjecture is true. These 3 expressions do generate Pythagorean triples. Now we are going to pick random numbers and plug them into the Euclidean method. Amanda Magli Page 30
31 Table. Examples of Triples Generated Using the Euclidean Method u v x y z In Table, column 1 and are the numbers being substituted for u and v. We picked a random number for u and a random number for v to be substituted into the Euclidean method. We then found out what x is by substituting the u and v numbers into the formula u v. We found out what y is by substituting in the u and v numbers for the formula uv. Finally we found out what z is by substituting in the u and v numbers into the formula u + v. Here is an example of what is explained above. Let u = 3 Let v = We substitute these numbers into the x formula 3  = 5 We substitute these numbers into the y formula (3)() = 1 Finally we substitute these numbers into the z formula Amanda Magli Page 31
32 3 + = 13 Now we are going to put these 3 numbers, 5, 1, 13 into the Pythagorean Theorem = 13 When we solve this equation, we see that the left side equals the right side so therefore 5, 1, 13 is a Pythagorean triple. Now we are going to look at the product of x, y and z Table 3. Product of x, y, and z x y z Product Table 3 shows the product of x, y and z. If we look at all of the products of x,y,z, notice that each answer ends with a 0. This means that each of the products is divisible by 10. Let s see if any other numbers divide into all of the products. The numbers and 5 have to go into all the products since they all end in a zero. The number 3 also goes into each of the products. We know this by adding the digits in each product together. If that answer is divisible by 3 then the whole product is divisible by 3. The numbers 4 and 6 also can be divided into each of the products. We know that the number 6 can be divided into each of the product since the numbers 3 and can be divided into each of the products. We know 4 can be divided into each of the products because if we Amanda Magli Page 3
33 look at the last two digits of each product, if that number is divisible by 4 then the whole product is divisible by 4. If there are double zeros as the last two numbers of the product, then look to the last 3 digits of the product and if 4 is divisible into that then the whole product is divisible by 4. We can conjecture by knowing that all of these numbers can be divided into each of the products, that 60 can also be divided into each of the products. RECOMMENDATIONS FOR FURTHER RESEARCH We generated 3 formulas to use to create Pythagorean Triples. One formula was called the Euclidean method, another formula was created when the smallest leg was odd, and a 3 rd formula was created when the smallest leg was even. There are some questions that still have to be answered. How many Pythagorean Triples are there? Are these the only 3 sets of formulas known that generate Pythagorean Triples? What patterns are there in the triples we find? Is the product of the elements of any Pythagorean Triple divisible by 60? Amanda Magli Page 33
34 BIBLIOGRAPHY Brown, S. A Surprsing Fact about Pythagorean Triples. Mathematics Teacher, Vol.78, No.8, (October 1985) pp Gerver, R. Writing Math Research Papers Berkeley, Ca: Key Curriculum Press Studyworks Mathematics Deluxe. CDROM. Boston: MathSoft, 00. Tirman, A. Pythagorean Triples. Mathematics Teacher, Vol. 79, No.9, (November 1986) Pp Amanda Magli Page 34
35 Proving the Product of Any Pythagorean Triple is Divisible by 60 Now we are going to try to prove the pattern that all Pythagorean triples are divisible by 60. We are going to take the 3 formulas found by the Euclidean method and let x = u v y = uv z = u + v We then are going to multiply all these formulas together and see if it is divisible by 60. (u v )(uv)( u + v ) Since is a factor, it must only be shown that (u v )uv( u + v ) We are going to let n = (v u )(uv)(u + v ) This is therefore divisible by 30 since we cut the 60 in half after we factored out the. If either u or v is divisible by, then n is divisible by. If u and v are odd, then so are u and v. For example if v = 5 u = 3 Then v = 5 u = 9 Therefore v  u is the difference of two odd numbers which gives us an even number. Amanda Magli Page 35
36 5 9 = 16 This proves that n is even and therefore is divisible by. This also must be divisible by 3 and 5 since the whole thing is divisible by 30. Divisibility by 3 We now are going to show that every positive integer is divisible by 3 or leaves a remainder of 1 or after it is divided by 3. Table 4: Integers Expressed as 3k, 3k+1, and 3k+ Positive Integer 3k 3k+1 3k As seen in Table 4, every integer can be expressed as 3k, 3k+1 or 3k+. In the next table, we show that every perfect square can be expressed as 3m or 3m+1. Amanda Magli Page 36
37 Table 5: Perfect Squares Expressed as 3m or 3m+1 Perfect Square 3m 3m Table 5 demonstrates the fact that every perfect square can be expressed as 3m or 3m+1. To prove that any perfect square can be expressed as this, we began with the fact that an integer can be written as 3k, 3k+1, or 3k+. Lets say Then b would be If we factor a 3 out of this number we get This represents 3 times a number or 3m. Now let Square Factor out a 3 b = 3k b = 9k 3(3k ) b = 3k+1 b = 9k + 6k + 1 3(3k +k) + 1 Amanda Magli Page 37
38 This represents 3m +1. Next we let b = 3k+ If we square this we get b = 9k + 1k + 4 Change the 4 into a Factor out a 3 3(3k + 4k + 1) + 1 This represents 3m + 1. This proves that a perfect square can always be expressed as either 3m or 3m+1. We now are going to introduce a new notation which will be used in the next few steps. This new notation is the divisibility notation which is represented by a vertical line and means divides evenly into. Here are some examples demonstrating the use of this new notation These examples show that 6 divides evenly into 18, 5 divides evenly into 10, but 10 does not divide evenly into 7 and 10 does not divide evenly into 5. We now are going to use this notation in showing that the expression (v u )uv( u + v ), is divisible by 3. Amanda Magli Page 38
39 If u or v can be written as 3m, then it is a multiple of 3 and so is its square root (u or v). If 3 u 3 u This shows that if 3 can divide evenly into u then 3 can divide evenly into u. An example is if 3 9 Then 3 3 If 3 uv, then 3 n. If 3 u and 3 v, let (v u )uv( u + v ) v = 3k+1 u = 3b+1 We plug it into the formula v u v u = (3k+1) (3b+1) Distribute the sign 3k 3b Factor a 3 out 3(kb) Therefore since 3 (v u ), 3 n. Amanda Magli Page 39
40 From this we conclude that (u v )uv( u + v ) must be divisible by 3. Now since we found out that this expression is divisible by and 3, we are going to prove that this expression is also divisible by 5. Divisibility by 5 Table 6 shows that every positive integer can be expressed as 5k, 5k+1, 5k+, 5k+3 or 5k+4. Table 6: Positive Integers Can Be Expressed as 5k, 5k+1, 5k+, 5k+3 or 5k+4 Positive 5k 5k+1 5k+ 5k+3 5k+4 Integer Table 6 expresses positive integers as 5k,5k+1, 5k+, 5k+3, or 5k+4. Every positive integer can be expressed this way. For example, the positive integer 3 can be expressed as 5k + 3 This is because 5 times 0 equals 0 and is 3. Amanda Magli Page 40
41 Next we are going to show that every perfect square can be expressed as 5m, 5m+1, or 5m1. Table 7: Every Perfect Square Can Be Expressed as 5m, 5m+ 1 or 5m1 Perfect Square 5m 5m+1 5m As seen in Table 7, every perfect square can be expressed as 5m, 5m+1 or 5m1. To prove that this is true we begin with the fact that any integer can be written as 5k, 5k+1, 5k+, 5k+3, or 5k+4. Let b = 5k Square both sides b = 5k Factor out a 5 5(5k ) This can be represented as 5 times m where m represents any positive integer. Now let b = 5k+1 Square both sides Amanda Magli Page 41
42 b = 5k +10k +1 Factor out a 5 5(5k +k) +1 This represents 5m +1. Next we let b = 5k+ Square both sides b = 5k +0k +4 Factor out a 5 5(5k +4k +1) 1 This represents 5m 1. Let b = 5k+3 Square both sides b = 5k +30k + 9 Express 9 as (101) b = 5k + 30k + (101) Use the associative property b = (5k + 30k + 10)  1 Factor out a 5 5(5k +6k + ) 1 This represents 5m 1. Finally we let Amanda Magli Page 4
43 b = 5k+4 We square both sides b = 5k +40k +16 Factor out a 5 5(5k + 8k + 3) + 1 This represents 5m + 1. This proves that a perfect square can be expressed as 5m, 5m+1 or 5m1. If u or v is a multiple of 5, then u or v is a multiple of 5 and so is the product of n. Therefore 5 n where n = (u v )uv( u + v ) If neither u or v is a multiple of 5 four other cases must be considered. One case is when v = 5k + 1 u = 5b + 1 If we plug it into the v u formula we get v u = (5k+1) (5b + 1) Distribute the sign 5k 5b Factor out a 5 5(kb) If 5 (v u ) then 5 divides evenly into the whole expression (u v )uv( u + v ) The second case that must be considered is when Amanda Magli Page 43
44 v = 5k1 u = 5b 1 If we plug it into the formula v u we get v u = (5k1) (5b 1) Distribute the sign 5k 5b Factor out a 5 5 (k b) Again 5 is a factor of n. The 3 rd case is when u = 5b + 1 v = 5k1 We plug it into the formula u + v this time u + v = (5b+1) + (5k1) This equals 5b + 5k Factor out a 5 5(b + k) Again this shows that if 5 (u + v ) then 5 n. The last case is when u = 5b 1 v = 5k + 1 We plug it into the formula u + v u + v = (5b 1) + (5k + 1) This equals Amanda Magli Page 44
45 5b + 5k Factor out a 5 5(b + k) Again this shows that since 5 u + v, 5 n. Thus n is divisible by, 3, and 5. Therefore this shows that the product of a Pythagorean triple, a, b,c, is always divisible by abc Amanda Magli Page 45
46 The next question that we will be answering is if there an infinite number of Pythagorean Triples? We are going to prove this by using indirect reasoning. Indirect reasoning is when we prove that something can not happen and therefore the other choice has to be correct. We are going to use the formulas found by the Euclidean method to complete this proof. x = u v y = uv z = u + v There are an infinite number of positive integers to choose from for u and v. We are trying to see by doing this proof, if from this infinite numbers that can be chosen for u and v, if two different sets of numbers will produce the same triple. Therefore our first step is that we are going to assume that there are a finite number of triples, which means that two different sets of numbers produce the same triple. Therefore u 1 u v 1 v and they form the same triple. Let x = u 1 v 1 x = u v Let y = u 1 v 1 Let y = u 1 v Amanda Magli Page 46
47 z = u 1 + v 1 z = u + v From the two expressions for x we can say u 1 v 1 = u v From the two expressions for z we can say u 1 + v 1 = u +v We can say this because the formulas above both equal z. If we add these two equations together we get u 1 = u Divide by u 1 = u This contradicts what we said first of all because we said that u 1 and u did not equal each other. Therefore from this, we conclude that our assumption, that there are a finite numbers of triples, is wrong and therefore the only other possible answer to this proof is that there are an infinite number of triples. Amanda Magli Page 47
48 RECOMMENDATIONS FOR FURTHER RESEARCH We proved that the product of any Pythagorean Triple is divisible by 60. There is still one question that is left that remains unanswered. If prime number divides evenly into a perfect square, then a prime number divides evenly into its square root. Amanda Magli Page 48
49 BIBLIOGRAPHY Brown, S. A Surprsing Fact about Pythagorean Triples. Mathematics Teacher, Vol.78, No.8, (October 1985) pp Gerver, R. Writing Math Research Papers Berkeley, Ca: Key Curriculum Press Studyworks Mathematics Deluxe. CDROM. Boston: MathSoft, 00. Tirman, A. Pythagorean Triples. Mathematics Teacher, Vol. 79, No.9, (November 1986) Pp Amanda Magli Page 49
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