Let
p
>
3
p>3
be a prime. Euler numbers
E
p
−
3
E_{p-3}
first appeared in H. S. Vandiver’s work (1940) in connection with the first case of Fermat’s Last Theorem. Vandiver proved that if
x
p
+
y
p
=
z
p
x^p+y^p=z^p
has a solution for integers
x
,
y
,
z
x,y,z
with
gcd
(
x
y
z
,
p
)
=
1
\gcd (xyz,p)=1
, then it must be that
E
p
−
3
≡
0
(
mod
p
)
E_{p-3}\equiv 0\,(\bmod \,p)
. Numerous combinatorial congruences recently obtained by Z.-W. Sun and Z.-H. Sun involve the Euler numbers
E
p
−
3
E_{p-3}
. This gives a new significance to the primes
p
p
for which
E
p
−
3
≡
0
(
mod
p
)
E_{p-3}\equiv 0\,(\bmod \,p)
.
For the computation of residues of Euler numbers
E
p
−
3
E_{p-3}
modulo a prime
p
p
, we use a congruence which runs significantly faster than other known congruences involving
E
p
−
3
E_{p-3}
. Applying this, congruence, via a computation in Mathematica 8, shows that there are only three primes less than
10
7
10^7
that satisfy the condition
E
p
−
3
≡
0
(
mod
p
)
E_{p-3}\equiv 0\,(\bmod \,p)
(these primes are 149, 241 and 2946901). By using related computational results and statistical considerations similar to those used for Wieferich, Fibonacci-Wieferich and Wolstenholme primes, we conjecture that there are infinitely many primes
p
p
such that
E
p
−
3
≡
0
(
mod
p
)
E_{p-3}\equiv 0\,(\bmod \,p)
.