Gilbreath's conjecture
From Wikipedia, the free encyclopedia
Norman Gilbreath's conjecture is a conjecture in number theory about the effect of difference operators on the sequence of prime numbers, attributed to Norman L. Gilbreath, in 1958.
[edit] Problem definition
Write down all the prime numbers, thus:
- 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...
and then write down the absolute difference of subsequent values in the above sequence, and then do the same with the resulting sequence. What you get looks like:
- 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...
- 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, ...
- 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, ...
Equivalently, let an be a value of the original sequence, and bn be a value of the new sequence; then
- bn = | an β an + 1 | .
Norman Gilbreath's conjecture states that the first value of this sequence always equals 1, except in the original sequence of primes. It has been verified for primes up to 1013[1].
[edit] Notes
- ^ A. M. Odlyzko, "Iterated absolute values of differences of consecutive primes," Mathematics of Computation, 61 (1993) pp. 373β380. [1]
 |
This number theory-related article is a stub. You can help Wikipedia by expanding it. |
