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Question: 18 a prove that for every exponent k 2 1...

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1.8. (a) Prove that for every exponent k 2 1, the congruence 2.2 + 1 0 (mod 5k) has a solution ck Z/5*z (b) Prove that the solutions in (a) can be chosen to satisfy rk+1 x (mod 5) for every k 2 1. (c) Prove that if we require the list of solutions a1, z2,rs,. .. to satisfy (b), then there are exactly two lists of solutions, the first being characterized by 2 (mod 5) and the second by a 3 (mod 5). Hint. Use induction on k. (This problem says that the equation z2 +1-0 has exactly two solutions in the 5-adic numbers. It is a special case of Hensels lemma.)
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