Gilbreath's conjecture is a strange hypothesis concerning prime numbers which was first suggested in 1958 by the American mathematician and amateur magician Norman L. Gilbreath following some doodlings on a napkin. Gilbreath started by writing down the first few primes:

 

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...

 

Under these he put their differences:

 

1, 2, 2, 4, 2, 4, 2, 4, 6, 2, ...

 

Under these he put the unsigned difference of the differences:

 

1, 0, 2, 2, 2, 2, 2, 2, 4, ...

 

And he continued this process of finding iterated differences:

 

1, 2, 0, 0, 0, 0, 0, 2, ...
1, 2, 0, 0, 0, 0, 2, ...
1, 2, 0, 0, 0, 2, ...
1, 2, 0, 0, 2, ...
1, 2, 0, 2, ...
1, 2, 2, ...
1, 0, ...
1, ...

 

Gilbreath's conjecture is that, after the initial two rows, the numbers in the first column are all one. No exception has been found to date, despite searches out to several hundred billions rows, and the conjecture is generally assumed to be true. However, it may have nothing to do with primes as such. The English mathematician Hallard Croft has suggested the conjecture may apply to any sequence that begins with 2 and is followed by odd numbers that increase at a "reasonable" rate and with gaps of "reasonable" size. If this is the case, Gilbreath's conjecture may not be as mysterious as it first seems, though it may be very difficult to prove.