Gilbreath's conjecture

Gilbreath's conjecture is a conjecture in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the results unsigned, and then repeating this process on consecutive terms in the resulting sequence, and so forth. The statement is named after mathematician Norman L. Gilbreath who, in 1958, presented it to the mathematical community after observing the pattern by chance while doing arithmetic on a napkin. In 1878, eighty years before Gilbreath's discovery, François Proth had, however, published the same observations along with an attempted proof, which was later shown to be false.

Gilbreath observed a pattern while playing with the ordered sequence of prime numbers

Computing the absolute value of the difference between term n+1 and term n in this sequence yields the sequence

If the same calculation is done for the terms in this new sequence, and the sequence that is the outcome of this process, and again ad infinitum for each sequence that is the output of such a calculation, the following five sequences in this list are

What Gilbreath—and François Proth before him—noticed is that the first term in each series of differences appears to be 1.

Stating Gilbreath's observation formally is significantly easier to do after devising a notation for the sequences in the previous section. Toward this end, let ${\displaystyle \{p_{n}\}}$ denote the ordered sequence of prime numbers ${\displaystyle p_{n}}$, and define each term in the sequence ${\displaystyle \{d_{n}\}}$ by

...
Wikipedia