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In this note, for any integers  m, n  ≥ 2, we find a condition on a positive integer  c  under which there exists a monic polynomialf  ∈  ℤ  [  x  ]of degree  n  for which  f ( x )   m    –  c  has  mn  integral roots counting with multiplicities. This is the case if and only if  m  = 2 and  c  is a constant that comes from a solution of the Prouhet-Tarry-Escott problem of size  n . For example, the smallest positive integer  c  for which there exists a monic degree 7 polynomialf  ∈  ℤ  [  x  ]such that  f ( x ) 2 – c  has 14 integral roots is  c  = 6620176679276160000.

Shifted Power of a Polynomial with Integral Roots