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Fermat Polynomial

The Polynomials obtained by setting $p(x)=3x$ and $q(x)=-2$ in the Lucas Polynomial Sequences. The first few Fermat polynomials are

$\displaystyle {\mathcal F}_1(x)$ $\textstyle =$ $\displaystyle 1$  
$\displaystyle {\mathcal F}_2(x)$ $\textstyle =$ $\displaystyle 3x$  
$\displaystyle {\mathcal F}_3(x)$ $\textstyle =$ $\displaystyle 9x^2-2$  
$\displaystyle {\mathcal F}_4(x)$ $\textstyle =$ $\displaystyle 27x^3-12x$  
$\displaystyle {\mathcal F}_5(x)$ $\textstyle =$ $\displaystyle 81x^4-54x^2+4,$  

and the first few Fermat-Lucas polynomials are
$\displaystyle f_1(x)$ $\textstyle =$ $\displaystyle 3x$  
$\displaystyle f_2(x)$ $\textstyle =$ $\displaystyle 9x^2-4$  
$\displaystyle f_3(x)$ $\textstyle =$ $\displaystyle 27x^3-18x$  
$\displaystyle f_4(x)$ $\textstyle =$ $\displaystyle 81x^4-72x^2+8$  
$\displaystyle f_5(x)$ $\textstyle =$ $\displaystyle 243x^5-270x^3+60x.$  

Fermat and Fermat-Lucas Polynomials satisfy

{\mathcal F}_n(1)={\mathcal F}_n


where ${\mathcal F}_n$ are Fermat Numbers and $f_n$ are Fermat-Lucas Numbers.

© 1996-9 Eric W. Weisstein