Find all pairs of integers ( x , m ) for which x − 2 3 + m 3 + − x − 2 3 + m 3 = 2

Question

Find all pairs of integers   (  x  ,  m  ) for which                              x          −          2                3            +      m        3    +            −                        x          −          2                3            +      m        3    =  2

Campasenabs · Accepted Answer

Let   y  =            x      −      2        3  , then we have  2  =            m      +      y        3    +            m      −      y        3                          |        3    8  =  m  +  y  +  3                    m        2            −              y        2              3    ⋅  2  +  m  −  yand thus  4  −  m  =  3                    m        2            −              y        2              3  so  64  −  48  m  +  12      m    2    −      m    3    =  27      m    2    −  27      y    2  or  27      y    2    =      m    3    +  15      m    2    +  48  m  −  64and finally                              27                      y            2                    =          (          m          −          1          )          (          m          +          8                      )            2                              Since   gcd  (  m  −  1  ,  m  +  8  )  ∣  9 we have   3  ∣  m  +  8 and   3  ∣  m  −  1 and so   m  +  8  =  3  b and   m  −  1  =  3  a so      y    2    =  a      b    2      ⟹    a  =      c    2  for some   c  ∈      N  Now                               m          =          3                      c            2                    +          1                     and   y  =  b  c  =  c  (      c    2    +  3  ) so                               x          =          2          +                      c            3                    (                      c            2                    +          3                      )            3

Find all pairs of integers ( x , m ) for which <mroot> <mroot>

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