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The n-cubic Permutations

Source: Iran TST 2014, second exam, day 2 ,problem 1

January 1, 2015

quadraticsnumber theory unsolvednumber theory

Problem Statement

n

is a natural number. We shall call a permutation

a_1,\dots,a_n

of

1,\dots,n

a quadratic(cubic) permutation if

\forall 1\leq i \leq n-1

we have

a_ia_{i+1}+1

is a perfect square(cube).

(a)

Prove that for infinitely many natural numbers

n

there exists a quadratic permutation.

(b)

Prove that for no natural number

n

exists a cubic permutation.

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