Prove or disprove: For all positive integers n , n 3 + n + 1 3 are irrational numbers.

Question

Prove or disprove: For all positive integers   n ,       n    3    +            n      +      1        3   are irrational numbers.

Hailee West · Accepted Answer

Note that   (  x  +  y      )          3        =      x          3        +      y          3        +  3  x  y  (  x  +  y  )  .. If   n or   n  +  1 is the cube of an integer, the result is clear by the uniqueness of prime factorization, so we assume that neither is the cube of an integer. Then neither is   n  (  n  +  1  ). Set   x  =      n                  1        3            and   y  =  (  n  +  1      )                  1        3              .. If   x  +  y is rational, then so are   (  x  +  y      )          3       and   3  x  y  (  x  +  y  ). Hence   3  x  y is rational. Thus   27  n  (  n  +  1  ) must be the cube of an integer, a contradiction.

Juan Lowe · Accepted Answer

Hint   If     n  ,  m  ∈      N   then cubing         n    3    +      m    3    ∈      Q      ⇒    n  m  ∈            N        3       (  ⇒     n  ,  m  ∈            N        3       i  f     (  n  ,  m  )  =  1  )

Prove or disprove: For all positive integers n , root(3)(n)+root(3)(n+1) are irrational numbers.

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