Prove 2 1 / 3 is irrational.

Question

Prove       2          1              /            3       is irrational.

relientaaho2 · Accepted Answer

Suppose       2                  1        3              =      n    m  . Then  2      m    3    =      n    3    ,or in other words      m    3    +      m    3    =      n    3    .But this contradicts Fermats Last Theorem.

starbright49ly · Accepted Answer

Suppose       2          1              /            3       is rational. Then   2  ⋅      m    3    =      n    3   for some   m  ,  n  ∈      N   . Looking at the left side, the power of two in the prime factorization of   2  ⋅      m    3   is of the form   3  k  +  1. On the right side, it must be of the form   3  l. This is a contradiction, because the factorizations on both sides must be the same by the fundamental theorem of arithmetic. Thus       2          1              /            3       cannot be rational.

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