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 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 incorrect.^[1]

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 following five sequences in this list are

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, and define each term in the sequence $(d_{n}^{1})$ by

d_{n}^{1}=p_{n+1}-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+1}^{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 equal to 1.

Verification and attempted proofs

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 equal to 1 for $k\leq n=3.4\times 10^{11}$ 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 0s and 2s for the next n numbers. This implies that the next n rows begin with a 1.

Generalizations

In 1980, Martin Gardner published a conjecture by Hallard Croft that stated that the property of Gilbreath's conjecture (having a 1 in the first term of each difference sequence) should hold more generally for every sequence that begins with 2, subsequently contains only odd numbers, and has a sufficiently low bound on the gaps between consecutive elements in the sequence.^[3] This conjecture has also been repeated by later authors.^[4]^[5] However, it is false: for every initial subsequence of 2 and odd numbers, and every non-constant growth rate, there is a continuation of the subsequence by odd numbers whose gaps obey the growth rate but whose difference sequences fail to begin with 1 infinitely often.^[6] Odlyzko (1993) is more careful, writing of certain heuristic reasons for believing Gilbreath's conjecture that "the arguments above apply to many other sequences in which the first element is a 1, the others even, and where the gaps between consecutive elements are not too large and are sufficiently random."^[2]^[7] However, he does not give a formal definition of what "sufficiently random" means.

References

^ ^a ^b Caldwell, Chris. "The Prime Glossary: Gilbreath's conjecture". The Prime Pages. Archived from the original on 2012-03-24. Retrieved 2008-03-07..
^ ^a ^b Odlyzko, A. M. (1993). "Iterated absolute values of differences of consecutive primes". Mathematics of Computation. 61 (203): 373–380. doi:10.2307/2152962. JSTOR 2152962. Zbl 0781.11037. Archived from the original on 2011-09-27. Retrieved 2006-05-25..
^ Gardner, Martin (December 1980). "Patterns in primes are a clue to the strong law of small numbers". Mathematical Games. Scientific American. Vol. 243, no. 6. pp. 18–28.
^ Guy, Richard K. (2004). Unsolved Problems in Number Theory. Problem Books in Mathematics (3rd ed.). Springer-Verlag. p. 42. ISBN 0-387-20860-7. Zbl 1058.11001.
^ Darling, David (2004). "Gilbreath's conjecture". The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes. John Wiley & Sons. pp. 133–134. ISBN 9780471667001. Archived from the original on 2016-05-05. Retrieved 2015-04-21.
^ Eppstein, David (February 20, 2011). "Anti-Gilbreath sequences". 11011110. Archived from the original on April 12, 2017. Retrieved April 12, 2017.
^ Chase, Zachary (2023). "A random analogue of Gilbreath's conjecture". Math. Ann. arXiv:2005.00530. doi:10.1007/s00208-023-02579-w.