Prove that for any integer n > 1, sqrt(n-1) + sqrt(n+1) is never rational.
Simon YAU
Number Theory
2011-03-12 7:54 am
回答 (2)
2011-03-12 7:15 pm
✔ 最佳答案
If √(n - 1) + √(n + 1) is rational, then [√(n - 1) + √(n + 1)]^2 = 2n + 2√(n^2 - 1) is also rational. However, it is impossible that √(n^2 - 1) is rational for any integer n > 1. This contradicts our assumption and we conclude that √(n - 1) + √(n + 1). is irrational for any integer n > 1 
2011-03-12 8:44 pm
Regrading the statement "However, it is impossible that √(n^2 - 1) is rational for any integer n > 1", the proof is given as follows:
Assume n^2 - 1 is a square number, for n>1,
that is, n^2 - 1 = k^2.
Rearranging yields (n-k)(n+k)=1,
and the result follows with n=1.
Q.E.D.
Assume n^2 - 1 is a square number, for n>1,
that is, n^2 - 1 = k^2.
Rearranging yields (n-k)(n+k)=1,
and the result follows with n=1.
Q.E.D.
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