Affiliation:
1. Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University , Naugarduko 24 , Vilnius , LITHUANIA
Abstract
ABSTRACT
In this note, for any integers m, n ≥ 2, we find a condition on a positive integer c under which there exists a monic polynomial
f
∈
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 polynomial
f
∈
ℤ
[
x
]
such that f(x)2–c has 14 integral roots is c = 6620176679276160000.
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