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 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 published the same observations.

Consider the prime numbers

Computing the absolute value of the difference between term ${\displaystyle n}$ and term ${\displaystyle n+1}$ 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.

Formally, let ${\displaystyle (p_{n})}$ denote the sequence of prime numbers. We can define the sequence ${\displaystyle (d_{n}^{k})}$ recursively by

Gilbreath's conjecture states that ${\displaystyle d_{1}^{k}=1}$ for all ${\displaystyle k\geq 1}$.

Several sources write that, as well as observing the pattern of Gilbreath's conjecture, François Proth released what he believed to be a proof of the statement that was later shown to be flawed. However, Zachary Chase disputes this, writing that although Proth called the observation a "theorem", there is no evidence that he published a proof, or false proof, of it.