WOLFRAM|DEMONSTRATIONS PROJECT

Gilbreath's Conjecture

​
k
min
1
k
max
10
n
max
5
1
2
3
4
5
1
1
2
2
4
2
2
1
0
2
2
2
3
1
2
0
0
0
4
1
2
0
0
0
5
1
2
0
0
0
6
1
2
0
0
2
7
1
2
0
2
0
8
1
2
2
2
2
9
1
0
0
0
0
10
1
0
0
0
0
A surprising conjecture about the gaps between primes, namely: Let
{
p
n
}
denote the ordered sequence of prime numbers
p
n
, and define each term in the sequence
{
d
{1,n}
}
by
d
{1,n}
=
p
n+1
−
p
n
,
where
n
is positive. Also, for each integer
k
greater than 1, let the terms in
{
d
{k,n}
}
be given by
d
{k,n}
=
d
{k-1,n+1}
-
d
{k-1,n}

.
Gilbreath's conjecture states that every term in the sequence
a
{k}
={
d
{k,1}
}
is 1. With this Demonstration you can check this amazing statement up to the
th
1000
difference series. The controls let you see the matrix of
d
{k,n}
, where
k
goes from
k
min
to
k
max
, and
n
goes from 1 to
n
max
. (If
k
min
>
k
max
, they switch roles. )