Abstract
The number of ideal solutions over $${\mathbb Z}_p$$ of the Tarry-Escott problem of degree two and the number of solutions over $${\mathbb Z}_p$$ of the related system of equations: $$\sum _{i=1}^{3} a_i^r = \sum _{i=1}^{3} b_i^r \quad \text {for}\quad r = 1,2,5$$ are completely determined.
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