Let
d
0
(
n
)
=
p
n
{d_0}(n) = {p_n}
, the nth prime, for
n
≥
1
n \geq 1
, and let
d
k
+
1
(
n
)
=
|
d
k
(
n
)
−
d
k
(
n
+
1
)
|
{d_{k + 1}}(n) = |{d_k}(n) - {d_k}(n + 1)|
for
k
≥
0
,
n
≥
1
k \geq 0,n \geq 1
. A well-known conjecture, usually ascribed to Gilbreath but actually due to Proth in the 19th century, says that
d
k
(
1
)
=
1
{d_k}(1) = 1
for all
k
≥
1
k \geq 1
. This paper reports on a computation that verified this conjecture for
k
≤
π
(
10
13
)
≈
3
×
10
11
k \leq \pi ({10^{13}}) \approx 3 \times {10^{11}}
. It also discusses the evidence and the heuristics about this conjecture. It is very likely that similar conjectures are also valid for many other integer sequences.