Wendt’s binomial circulant determinant,
W
m
{W_m}
, is the determinant of an m by m circulant matrix of integers, with (i, j)th entry
(
m
|
i
−
j
|
)
\left ( {\begin {array}{*{20}{c}} m \\ {|i - j|} \\ \end {array} } \right )
whenever 2 divides m but 3 does not. We explain how we found the prime factors of
W
m
{W_m}
for each even
m
≤
200
m \leq 200
by implementing a new method for computations in algebraic number fields that uses only modular arithmetic. As a consequence we prove that if p and
q
=
m
p
+
1
q = mp + 1
are odd primes, 3 does not divide m, and
m
≤
200
m \leq 200
, then the first case of Fermat’s Last Theorem is true for exponent p.