The IthFermat(i) command returns the ith Fermat number.

Fermat numbers are of the form $2^{2^i}+1$ where $i$ is a non-negative integer.

The output option specifies the format of the returned solution. This option can be an equation of the form output=format or output=list, where format can be any one of the following:

value : Returns the value of the ith Fermat number. This is the default output format.

prime : Returns true, false, or FAIL, depending on whether the primality of the ith Fermat is known.

completelyfactored : Returns true, false or FAIL depending on whether the ith fermat has been completely factored or not. If FAIL is returned, the character of the ith Fermat is unknown.

factors : Returns the prime factors for the ith Fermat number, if they are known. If the ith Fermat number is prime, [] is returned. Otherwise, the factors are unknown.

record : Returns a record containing all of the above output options. Each of the outputs can be queried using :-outputname.

$\mathrm{with}\left(\mathrm{NumberTheory}\right)\:$

$\mathrm{IthFermat}\left(0\right)$

$3$

$2^{2^0}+1$

$3$

$\mathrm{seq}\left(\mathrm{IthFermat}\left(i\right)\,i=0..4\right)$

$3,5,17,257,65537$

$\mathrm{IthFermat}\left(1\,\mathrm{output}=\mathrm{prime}\right)$

$\mathrm{true}$

$\mathrm{IthFermat}\left(3\,\mathrm{output}=\left[\mathrm{prime}\,\mathrm{value}\,\mathrm{completelyfactored}\right]\right)$

$\mathrm{true},257,\mathrm{true}$

$\mathrm{F\_\_5}≔\mathrm{IthFermat}\left(5\,\mathrm{output}=\mathrm{record}\right)$

$\mathrm{F\_\_5}≔\mathrm{Record}\left(\mathrm{value}=4294967297\,\mathrm{prime}=\mathrm{false}\,\mathrm{completelyfactored}=\mathrm{true}\,\mathrm{factors}=\left[\left({\left(2\right)}^7\left(5\right)+1\right)\left({\left(2\right)}^7\left(3\right)\left(17449\right)+1\right)\right]\right)$

$\mathrm{F\_\_5}:-\mathrm{value}$

$4294967297$

$\mathrm{F\_\_5}:-\mathrm{prime}$

$\mathrm{false}$

$\mathrm{F\_\_5}:-\mathrm{factors}$

$\left[\left({\left(2\right)}^7\left(5\right)+1\right)\left({\left(2\right)}^7\left(3\right)\left(17449\right)+1\right)\right]$

$f≔\mathrm{IthFermat}\left(49093\,'\mathrm{output}'='\mathrm{factors}'\right)$

$f≔\left[\left(3\right)\left(5\right)\left(11\right){\left(2\right)}^{49095}+1\right]$

$f≔\mathrm{expand}\left(f\left[1\right]\right)\:$

$2\&ˆ\left(2^{49093}\right)+1\mathbf{mod}f$

$0$

$\mathrm{FermatDF}≔n\mapsto \mathrm{DataFrame}\left(⟨⟨\mathrm{seq}\left(2^{{\mathrm{`2`}}^i}+1\,i\mathbf{in}n\right)⟩\left|⟨\mathrm{seq}\left(\mathrm{length}\left(\mathrm{IthFermat}\left(i\right)\right)\,i\mathbf{in}n\right)⟩\right|⟨\mathrm{seq}\left(\mathrm{IthFermat}\left(i\,\mathrm{output}=\mathrm{prime}\right)\,i\mathbf{in}n\right)⟩|⟨\mathrm{seq}\left(\mathrm{IthFermat}\left(i\,\mathrm{output}=\mathrm{completelyfactored}\right)\,i\mathbf{in}n\right)⟩⟩\,\mathrm{rows}=\left[\mathrm{op}\left(n\right)\right]\,\mathrm{columns}=\left[\mathrm{F\_\_n}\,\mathrm{`\#\; of\; digits`}\,\mathrm{Prime}\,\mathrm{`Completely\; Factored`}\right]\right)\:$

$\mathrm{FermatDF}\left(\left[\mathrm{seq}\left(0..7\right)\right]\right)$

$\left[\begin{array}{ccccc}& \mathrm{F\_\_n}& \mathrm{\#\; of\; digits}& \mathrm{Prime}& \mathrm{Completely\; Factored}\\ 0& 3& 1& \mathrm{true}& \mathrm{true}\\ 1& 2^2+1& 1& \mathrm{true}& \mathrm{true}\\ 2& 2^{2^2}+1& 2& \mathrm{true}& \mathrm{true}\\ 3& 2^{2^3}+1& 3& \mathrm{true}& \mathrm{true}\\ 4& 2^{2^4}+1& 5& \mathrm{true}& \mathrm{true}\\ 5& 2^{2^5}+1& 10& \mathrm{false}& \mathrm{true}\\ 6& 2^{2^6}+1& 20& \mathrm{false}& \mathrm{true}\\ 7& 2^{2^7}+1& 39& \mathrm{false}& \mathrm{true}\end{array}\right]$

$\mathrm{FermatDF}\left(19\right)$

$\left[\begin{array}{ccccc}& \mathrm{F\_\_n}& \mathrm{\#\; of\; digits}& \mathrm{Prime}& \mathrm{Completely\; Factored}\\ 19& 2^{2^{19}}+1& 157827& \mathrm{false}& \mathrm{false}\end{array}\right]$

$\mathrm{GetRow}\left(\mathrm{FermatDF}\left(21\right)\,1\right)$

$\left[\begin{array}{cc}\mathrm{F\_\_n}& 2^{2^{21}}+1\\ \mathrm{\#\; of\; digits}& 631306\\ \mathrm{Prime}& \mathrm{false}\\ \mathrm{Completely\; Factored}& \mathrm{false}\end{array}\right]$

The NumberTheory[IthFermat] command was introduced in Maple 2017.

For more information on Maple 2017 changes, see Updates in Maple 2017.