Question

14. More Work with Pythagorean Triples. In Exercise (13), we verified that each of the following triples of natural numbers are Pythagorean triples: • 3, 4, and 5 • 6, 8, and 10 • 8, 15, and 17 • 10, 24, and 26 . 12, 35, and 37 • 14, 48, and 50 (a) Focus on the least even natural number in each of these Pythagorean triples. Let n be this even number and find m so that n = 2m. Now try to write formulas for the other two numbers in the Pythagorean triple in terms of m. For example, for 3, 4, and 5, n = 4 and m = 2, and for 8, 15, and 17, n = 8 and m = 4. Once you think you have formulas, test your results with m = 10. That is, check to see that you have a Pythagorean triple whose smallest even number is 20. (b) Write a proposition and then write a proof of the proposition. The proposition should be in the form: If m is a natural number and m > 2, then

Answer 1

@ for 3,4,5 we see that n=4 and so the the other m =2 triplets are 4-1 = 3 4+1 - 5 a = 2 -1 = as 2 tl = for 8, 15, 17 , n=8 , m= 4 Then a = 4 1 = 16-1 = 15 b= 4²+1 = 16+1 = 17 we will test our m=10 results with Then a = 10-12 100 - 99 I be lo'tl -100+1 = 101 and n=am = 20 Now verify for 20,99, 101 101? = 10201 20² +992= 400+ 9801 - 10201 verified that 1012 = 20² +99².

e a proposition : . and If n mi2 is then natural number am , mil, m²+1 are pythagorean triplets . proof: Now we will prove this (m?+) - m" tit 2m2 (m?-1) + (2m) - m'+1-2m2 + 4m? mi +2m2 til Hence (m’+1) = (m?-)? + 12m)2 Proved,