We estimate the least prime factor p of the binomial coefficient
(
k
N
)
\left ( {_k^N} \right )
for
k
≥
2
k \geq 2
. The conjecture that
p
≤
max
(
N
/
k
,
29
)
p \leq \max (N/k,29)
is supported by considerable numerical evidence. Call a binomial coefficient good if
p
>
k
p > k
. For
1
≤
i
≤
k
1 \leq i \leq k
write
N
−
k
+
i
=
a
i
b
i
N - k + i = {a_i}{b_i}
, where
b
i
{b_i}
contains just those prime factors
>
k
> k
, and define the deficiency of a good binomial coefficient as the number of i for which
b
i
=
1
{b_i} = 1
. Let
g
(
k
)
g(k)
be the least integer
N
>
k
+
1
N > k + 1
such that
(
k
N
)
\left ( {_k^N} \right )
is good. The bound
g
(
k
)
>
c
k
2
/
ln
k
g(k) > c{k^2}/\ln k
is proved. We conjecture that our list of 17 binomial coefficients with deficiency
>
1
> 1
is complete, and it seems that the number with deficiency 1 is finite. All
(
k
N
)
\left ( {_k^N} \right )
with positive deficiency and
k
≤
101
k \leq 101
are listed.