A296136 - OEIS

A296136

Smallest positive integer equal to the difference between two monic polynomials of degree n with integer zeros.

1, 1, 4, 36, 5040, 14400

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OFFSET

1,3

COMMENTS

The multisets of zeros of two degree-n polynomials with an integer difference form an ideal solution to the Prouhet-Tarry-Escott problem of size n. Correspondingly, a(n) equals the smallest difference between the products of multisets terms in an ideal solution. By the Newton-Girard formulas, the same ideal solutions deliver the smallest difference between the sums of n-th powers (cf. A296137).

LINKS

Table of n, a(n) for n=1..6.

Wikipedia, Prouhet-Tarry-Escott problem

EXAMPLE

a(1) = 1 = x - (x-1),

a(2) = 1 = (x-1)^2 - (x-2)*x,

a(3) = 4 = (x+1)*(x-2)^2 - x^2*(x-3), [corrected by Alex Meiburg, Dec 29 2017]

a(4) = 36 = x^2*(x-5)^2 - (x+1)*(x-2)*(x-3)*(x-6),

a(5) = 5040 = (x+8)*(x+4)*x*(x-8)*(x-9) - (x+7)*(x+6)*(x-2)*(x-6)*(x-10),

a(6) = 14400 = (x+7)^2*x^2*(x-7)^2 - (x+8)*(x+5)*(x+3)*(x-3)*(x-5)*(x-8).

CROSSREFS

Cf. A296137.