Solution Commentary: Solution of Main Problems: Pythagorean Triples 1. If m = (n + 1), n = [m -1]/ and n+1 = [m +1]/. Then, by substitution, the equation n + (n + 1) = (n+1) m + 1 becomes + m =. Now, because an even number squared is even and the expression (n+1) is odd, we know that both m and m are odd. Thus, for a short sequence of odd numbers, we produce these Pythagorean triples (increasing side length): m a b c 3 3 4 5 5 5 1 13 7 7 4 5 9 9 40 41 11 11 60 61 13 13 84 85 Though this process generates many Pythagorean triples (seemingly all primitive?), but not all Pythagorean triples. For example, the triple (8, 15, 17) will never be generated for two reasons. First, the lowest term is 8, not an odd number. And second, the difference between the two larger numbers in the triple is not 1. Thus, perhaps at best, this process generates all Pythagorean triples where the hypotenuse differs from one side by 1.. Let m be an odd number and one side of the right triangle. Then, by Pythagoras method, the other side is (m -1)/ numbers. To get the hypotenuse, add to get [(m -1)/] + 1 = (m +1)/. This is exactly the same m + 1 method, since + m =. 3. Let p be an even number and one side of the right triangle. Then, by Plato s method, the hypotenuse is [p/] +1 and the other side is [p/] -1. Thus, for a short sequence of even numbers, we produce these Pythagorean triples (increasing side length): p a b c 4 3 4 5 6 6 8 10 8 8 15 17 10 10 4 6 1 1 35 37 14 14 48 50
m 1 4. In Pythagorean s method, the two larger sides m + 1 and will always differ by 1, and thus have no common factor. However, in Plato s method which starts with an even side, the two sides [p/] -1 and [p/] +1 not only differ by but will also both be even whenever [p/] is odd. Extension 1: With a = p q, b = pq, and c = p + q, we have a + b = (p q ) + (pq) = p 4 p q + q 4 + 4p q = p 4 + p q + q 4 = (p + q ) = c. If p and q had a common factor d > 1, then d p and d q implies that d (p -q ), d (pq), and d (p +q ), thereby producing a non-primitive triple. p q a b c 1 3 4 5 3 1 8 6 10 3 5 1 13 4 1 15 8 17 4 3 7 4 5 5 1 4 10 6 5 1 0 9 5 3 16 30 34 5 4 9 40 41 6 1 35 1 37 6 5 11 60 61 7 1 48 14 50 7 45 8 53 7 3 40 4 58 7 4 33 56 65 7 5 4 70 74 7 6 13 84 85 Many non-primitive triples are still generated, such as (6,8,10), (10,16,4), (16,30,34), (14,48,50), (40,4,58), and (4,70,74). By comparison, this table encompasses all of the triples listed in the two previous tables. Nonetheless, some new Pythagorean triples are generated, such as (0,1,9), (8,45,53), and (33,56,65). NOTE: Sierpinski (196) proves that this set of expressions will generate all primitive Pythagorean triples. Extension : First, if x= 9, let p = 1 and q = 9. Then, a = 9 = 3, b = (9-1)/ = 4, and c = (9+1)/ = 5, producing the triple (3,4,5). And, if x= 5, let p = 1 and q = 5. Then, a = 5 = 5, b = (5-1)/ = 1, and c = (5+1)/ = 13, producing the triple (5,1,13). To prove a + b = c starting with a = pq, we can prove that a = b c = (b+c)(b-c). p + q p q p + q p q p q Thus, substituting, ( c b)( c a) = + = = pq = a. Note:
Extension 3: Using the conditions in Problem #1 and the right triangle s diagram, let m 1 AB = a = m, AC = b = m + 1, BC = c = where a + b = c. The area of the ab ar br cr triangle can be visualized in two ways, where A = = + +. Solving for r and ab simplifying, r a + b + c m 1 and simplifying algebraically, m r = =. Expanding the table + 1 + + m m generated in Problem #1, we obtain an interesting relationship, showing that the respective radii are the set of whole numbers: m a b c r 3 3 4 5 1 5 5 1 13 7 7 4 5 3 9 9 40 41 4 11 11 60 61 5 13 13 84 85 6 Open-Ended Exploration: This exploration can be quite productive, with many perhaps unknown discoveries yet to be made. Freitag (199), Bertucci (1991), and Horadam (1961) are some useful resources. Teacher Commentary: When exploring Pythagorean triples and trying to connect them back to the visual interpretation of proofs of the Pythagorean Theorem, students may want to explore the connection to the Greek idea of applying areas. As a first step, the square of side length c can be visualized as containing a square of side length a. Once the latter square is removed, is it possible to transform the remaining area into a square of side length b? You might want to have students explore Schwaller s (1979) interesting approach to this question, which ends up generating Pythagorean triples. In Problem #1, Pythagoras method produced some Pythagorean triples, but it seemed to miss some generated by Plato s method, such as (8,15,17). But, if m were not restricted to odd integers, other triples can be produced. For example, using m= in the expressions, generates the non-integer Pythagorean triple (4,7½,8½), which can be multiplied by to get the integer Pythagorean triple (8,15,17). If fact, if m is allowed to be a rational number, it can be shown that Pythagoras method would generate all of the Pythagorean triples. And, if m = p/q is substituted into Pythagoras expressions for generating triples, the more common set of expressions pq, p - q and p + q appears as explored in Extension Problem #1.
Extension Problem # explored Euclid s method for producing Pythagorean triples. This method has been analyzed and discussed often as to whether or not Euclid s method produces all of the triples. Laubenbacher & Pengelley (1999 pp. 17-178) document thoroughly that this is the case, though such may be deceptive. Boulger (1993) was the source for the solution shown in Extension Problem #3. However, the problem has been discussed in many references, and even solved in very different ways, such as in Maor (007). As a good source of writing projects, students can explore any of the following ideas relative to Pythagorean triples: Extension Problem #1 focused on the standard set of expressions for generating Pythagorean triples. Investigate and compare different approaches to deriving these expressions, such as the number theoretic approach (Ore, 1988), the geometric approach (Kleiman, 197), the trigonometric approach (Maiorano, 1995), or even the complex number approach (Kerins et al, 003). Though Pythagorean triples usually are viewed as a topic in number theory, they can be viewed visually. Investigate the visual aspects, using sources such as the visual patterns in Nelson (007), the geoboard approach in Bonsangue (1997), the 3-D graphs in Shabanian (1995), and the distribution patterns in Oliver (1971). Investigate other unusual properties of Pythagorean triples, such as: 1. The product of the members of every Pythagorean triple (a,b,c) is a multiple of 60. (Brown, 1985; Rudolph, 1987). The connection between Pythagorean triples and special triangles where all three sides and the area are integers, plus the area is an integer multiple of the perimeter. (Markov, 006) 3. For Pythagoras method, explore how his triangle s semiperimeters and areas are connected to triangular numbers and the sums of consecutive square numbers. (Maiorano, 1994) 4. Based on an interpretation of Plimpton-3 clay tablet (ca. 1500 B.C.), the Babylonians basically formed an infinite chain of Pythagorean triples (a,b,c) by using a rectangular rearrangement technique where a square with integral side length a visually becomes a rectangle with integral side lengths b and c, such that a = c - b = (c + b)(c - b). (Brudner, 1983) Additional References: Bertucci, J. (1991). From Fibonacci numbers to Pythagorean triples. Mathematics Teacher. May, p. 356. Bonsangue, M. (1997). A geometrical representation of primitive Pythagorean triples. Mathematics Teacher. Vol. 90#5. pp. 350-354.
Boulger, W. (1993). Triple your fun again. Mathematics Teacher. November, p. 704. Brown, S. (1985). A surprising fact about Pythagorean triples. Mathematics Teacher. October, pp. 540-541. Brudner, H. (1983). Babylonian doubles and Pythagorean triples. Educational Technology. March, pp. 34-36 and April, pp. 33-36. Freitag, H. (199). Generating triples. Mathematics Teacher. January, p. 10. Heath, T. (1956). The Thirteen Books of Euclid s Elements. Vol 1. Dover Publications. Horadam, A. (1961). Fibonacci number triples. American Mathematical Monthly. Vol. 68, pp. 751-753. Kerins, B. et al. (003). Gauss, Pythagoras, and Heron. Mathematics Teacher. Vol. 96#5. pp. 350-357. Kleiman, S. (197). Pythagorean triples by geometry. Two-Year College Mathematics Journal. Spring, pp. 39-41. Laubenbacher, R. and Pengelley, D. (1999). Mathematical Expeditions: Chronicles by the Explorers. Springer. Maiorana, P. (1994). More triples fun. Mathematics Teacher. November, p. 66. Maiorano, P. (1995). Triples by way of trigonometry. Mathematics Teacher. March, page****. Maor, E. (007). The Pythagorean Theorem: A 4000 Year History. Princeton University Press. Markov, L. (006). Pythagorean triples and the problem A = mp for triangles. Mathematics Magazine. Vol. 79#, pp. 114-11. Nelson, R. (007). Visual gems of number theory. Math Horizons, November, 7-9, 31. Oliver, B. (1971). The pattern of Pythagorean numbers. Mathematics Teacher. May, pp. 449-454. Ore, O. (1988). Number Theory and Its History. Dover Publications. Rudolph, M. (1987). Triple s surprise. Mathematics Teacher. May, pp. 343-344.
Schwaller, R. (1979). A method of generating Pythagorean triples. Int. J. Math. Educ. Sci. Tech. Vol. 10#1, pp. 75-77. Shabanian, S. (1995). Computer graphical modeling of Pythagorean triplets in 3-D space. Scientific Computing & Automation. June, pp. 45-48. Sierpinski, W. (196). Pythagorean Triangles. New York: Yeshiva University. pp. 5-7.