Gilbreath's conjecture

Gilbreath's conjecture is a hypothesis, or 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.^[1] 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.^[1]

1 Motivating arithmetic
2 The conjecture
3 Verification and attempted proofs
4 See also
5 References

Motivating arithmetic

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

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

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

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

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 first five sequences in this list are given by

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

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, ...

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

The conjecture

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 $\{p_{n}\}$ denote the ordered sequence of prime numbers $p_{n}$ , and define each term in the sequence $\{d_{n}\}$ by

$d_{n} = p_{n%2B1} - p_{n}$ ,

where $n$ is positive. Also, for each integer $k$ greater than 1, let the terms in $\{d_{n}^{k}\}$ be given by

$d_{n}^{k} = |d_{n%2B1}^{k-1}-d_{n}^{k-1}|$ .

Gilbreath's conjecture states that every term in the sequence $a_{k} = {d_{1}^{k}}$ for positive $k$ is 1.

Verification and attempted proofs

As of 2011^[update], no valid proof of the conjecture has been published. As mentioned in the introduction, François Proth released what he believed to be a proof of the statement that was later shown to be flawed. Andrew Odlyzko verified that $d_{1}^{k}$ is 1 for $k \leq n = 3.4\times10^{11}$ in 1993,^[2] but the conjecture remains an open problem. Instead of evaluating n rows, Odlyzko evaluated 635 rows and established that the 635th row started with a 1 and continued with only 0's and 2's for the next n numbers. This implied that the next n rows began with a 1.

References

^ ^a ^b Caldwell, Chris, "The Prime Glossary: Gilbreath's conjecture", The Prime Pages, http://primes.utm.edu/glossary/page.php?sort=GilbreathsConjecture .
^ Odlyzko, A. M. (1993), "Iterated absolute values of differences of consecutive primes", Mathematics of Computation 61: 373–380, http://www.dtc.umn.edu/~odlyzko/doc/arch/gilbreath.conj.ps .

Gilbreath's conjecture

Contents

Motivating arithmetic

The conjecture

Verification and attempted proofs

See also

References