Gilbreath's guess

Gilbreath's conjecture is an unproven number theoretic claim that concerns the prime numbers . It is attributed to Norman L. Gilbreath (* 1936) for 1958, he is said to have discovered it while scribbling on a napkin . The conjecture was published in 1878 by François Proth together with an alleged evidence that later turned out to be flawed.

You write the sequence of prime numbers in the first line : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, ... Then you calculate the absolute values ​​of the Differences between consecutive members and so note the second line. The third and all following lines are formed in the same way:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, ...

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

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

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

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

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

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

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

1, 2, 2, 2, 2, 2, 2, 2, ...

1, 0, 0, 0, 0, 0, 0,…

1, 0, 0, 0, 0, 0,…

Gilbreath's guess is that the first value of every row except the first row is 1. Andrew Odlyzko provided a check for the first approximately lines. ${\displaystyle 3\cdot 10^{11}}$

Web links

• Eric W. Weisstein : Gilbreath's Conjecture . In: MathWorld (English).

Individual evidence

1. ^ Gilbreath's conjecture . The Prime Glossary.
2. ^ François Proth: Sur la series des nombres premiers. In: Nouv. Corresp. Math. 4, 1878, pp. 236-240.
3. ↑ gilbreath.conj.ps .
4. ↑ Andrew M. Odlyzko: Iterated absolute values ​​of differences of consecutive primes . In: Mathematics of Computation . tape 61 , no. 203 , 1993, pp. 373-380 , doi : 10.1090 / S0025-5718-1993-1182247-7 .