Prove that 2022/n+4n is a perfect square iff 2022/n−8n is a perfect square

Charlie Conner 2022-10-06 Answered
Prove that 2022 n + 4 n is a perfect square iff 2022 n 8 n is a perfect square
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Answers (1)

omeopata25
Answered 2022-10-07 Author has 5 answers
You can greatly reduce the number of cases to check by working modulo 4. We have
2022 = 2 3 337 2 3 1 ( mod 4 ) .
Since a square must be 0 or 1 mod 4, we can immediately reduce to the two cases n { 6 , 2022 }. And of course n = 2022 is impossible since 2022 n 8 n will be negative.
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