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Define a sequence of positive integers by the rule that a(n) = n for 1 <= n <= 3, and for n >= 4, a(n) is the smallest number not already in the sequence which has a common factor with a(n-2) and is relatively prime to a(n-1). We show that this is a permutation of the positive integers. The remarkable graph of this sequence consists of runs of alternating even and odd numbers, interrupted by small downward spikes followed by large upward spikes, suggesting the eruption of geysers in Yellowstone National Park. On a larger scale the points appear to lie on infinitely many distinct curves. There are several unanswered questions concerning the locations of these spikes and the equations for these curves.
In this note, a criterion for a class of binomials to be permutation polynomials is proposed. As a consequence, many classes of binomial permutation polynomials and monomial complete permutation polynomials are obtained. The exponents in these monomials are of Niho type.
As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree $(q^n-1)/(q-1)$ of ${rm L}_n(q)$ is prime. We present heuristic a
In this paper, we present three classes of complete permutation monomials over finite fields of odd characteristic. Meanwhile, the compositional inverses of these complete permutation polynomials are also proposed.
Given a permutation polynomial of a large finite field, finding its inverse is usually a hard problem. Based on a piecewise interpolation formula, we construct the inverses of cyclotomic mapping permutation polynomials of arbitrary finite fields.
Let $f(X)=X(1+aX^{q(q-1)}+bX^{2(q-1)})inBbb F_{q^2}[X]$, where $a,binBbb F_{q^2}^*$. In a series of recent papers by several authors, sufficient conditions on $a$ and $b$ were found for $f$ to be a permutation polynomial (PP) of $Bbb F_{q^2}$ and, in