In this problem, you will prove that there are no positive integers a, b, c, and d such that
(a) Prove that for all integers m and n, if 3 | (m2 + n2) then 3 | m and 3 | n. (Hint: By exercise 14(c), either m ≡ 0 (mod 3) or m ≡ 1 (mod 3) or m ≡ 2 (mod 3), and also either n ≡ 0 (mod 3) or n ≡ 1 (mod 3) or n ≡ 2 (mod 3). This gives nine possibilities. Determine which of these possibilities are compatible with the assumption that 3 | (m2 + n2).)
Now suppose there are positive integers satisfying (∗). Let
Then S != ∅, so by the well-ordering principle we can let d be the smallest element of S. Let a, b, and c be positive integers satisfying (∗).
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(b) Prove that 3 | c and 3 | d. (Hint: Add the two equations in (∗) and then apply part (a).)
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(c) Prove that 3 | a and 3 | b. (Hint: Add the two equations in (∗) and then apply part (b).)
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(d) Show that there is an element of S that is smaller than d, which contradicts our choice of d. (Hint: Combine parts (b) and (c).)