Sylvester Cyclotomic Number

Author

Eric W. Weisstein
February 17, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/NumberTheory/SylvesterCyclotomicNumber.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/SylvesterCyclotomicNumber.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Code

SylvesterCyclotomicNumber[n_,{P_,Q_}]:=Module[{D=(P^2)-4Q,α,β,k},
{α,β}=(P+{1,-1}(Sqrt[D]))/2;
Product[(α-E^(2Pi I k/n)β)^KroneckerDelta[GCD[k,n],1],{k,n-1}]
]

U[n_,{P_,Q_}]:=Module[{k,D=(P^2)-4Q,α,β},
{α,β}=(P+{1,-1}(Sqrt[D]))/2;
((α^n)-(β^n))/(α-β)
]

Enumeration

Table[{n,SylvesterCyclotomicNumber[n,{P,Q}]//FullSimplify},{n,0,10}]//TableForm//TraditionalForm

Table[Times@@(SylvesterCyclotomicNumber[#,{P,Q}]&/@Divisors[n]),{n,0,10}]//FullSimplify//TableForm//TraditionalForm

Table[U[n,{P,Q}],{n,0,10}]//FullSimplify//TableForm//TraditionalForm