Given a Lucas sequence with parameters 
and 
, discriminant 
, and roots 
and 
, the Sylvester cyclotomic numbers are
 |
(1)
|
where
 |
(2)
|
is a primitive root of unity and the product is over all exponents 
relatively prime to 
such that 
.
For small 
,
the first few values are
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
These numbers satisfy
 |
(10)
|
where as usual 
.
Ward (1954) gave a primality test involving these numbers.
