Gilbreath's conjecture

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Gilbreath's conjecture is a conjecture in number theory about the effect of difference operators on the sequence of prime numbers. It is named after Norman L. Gilbreath who came up with it in 1958. Long before that François Proth had actually discovered and published this effect in 1878. Proth claimed to have proved it but the proof was not correct.^[1]

[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 (3-2=1; 5-3=2; 7-5=2; 11-7=4; etc.) 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 a_n be a value of the original sequence, and b_n be a value of the new sequence; then

b_n = | a_n − a_{n + 1} | .

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.^[2]

[edit] Notes

1. ^ Chris Caldwell, The Prime Glossary: Gilbreath's conjecture at The Prime Pages.
2. ^ A. M. Odlyzko, "Iterated absolute values of differences of consecutive primes", Mathematics of Computation, 61 (1993) pp. 373–380.