Consider the following theorem: There do not exist three consecutive odd integers a, b, and c such that a2 + b2 = c2.
(a) Complete the following restatement of the theorem: For every three consecutive odd integers a, b, and c, _____.
(b) Change the sentence in part (a) into an implication p ⇒ q: If a, b, and c are consecutive odd integers, then ______.
(c) Fill in the blanks in the following proof of the theorem.
Proof: Let a, b, and c be consecutive odd integers. Then a = 2k+ 1, b = ______, and c = 2k + 5 for some integer k. Suppose a2 + b2 = c2. Then (2k + 1)2 + (______ )2 = (2k + 5)2.
It follows that 8k2 + 16k + 10= 4k2 + 20k + 25 and 4k – 4k – _____ = 0. Thus k = 5 /2 or k =______ This contradicts k being an ______. Therefore, there do not exist three consecutive odd integers a, b, and c such that a2 + b2 = c2.
(d) Which of the tautologies in Example 1.3.12 best describes the structure of the proof?
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