Gilbreath's conjecture is a conjecture in number theory about the effect of difference operators on the sequence of prime numbers. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes In Mathematics, a difference operator maps a function, f ( x) to another function f ( x + a) &minus f ( x In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 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. François Proth (1852 - 1879 was a French self-taught mathematician farmer who lived near Verdun, France. Proth claimed to have proved it but the proof was not correct. [1]
Problem definition
Write down all the prime numbers, thus:
- 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, . In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 . .
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 an be a value of the original sequence, and bn be a value of the new sequence; then
- bn = | an − an + 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]
Notes
- ^ Chris Caldwell, The Prime Glossary: Gilbreath's conjecture at The Prime Pages. The Prime Pages is a website about Prime numbers maintained by Prof
- ^ A. M. Odlyzko, "Iterated absolute values of differences of consecutive primes", Mathematics of Computation, 61 (1993) pp. Andrew Odlyzko is a mathematician who is the head of the University of Minnesota 's Digital Technology Center 373–380.
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