This picture displays the process for the 64 first prime numbers:
{2,3,5,7,11,13,17,19,23,29,,31,37,41,43,47,53,59,61,67,71,,73,79,83,89,97,101,103,107,109,113,,127,131,137,139,149,151,157,163,167,173,,179,181,191,193,197,199,211,223,227,229,,233,239,241,251,257,263,269,271,277,281,,283,293,307,311}
-top row of the picture- with the following colors regarding the numbers:
0 = Dark Yellow, 1 = Cyan, 2 = Light Yellow,
when all other numbers -{3,5,7,11,...}- are Dark Red...
According to the the Gilbreath Conjecture the left-hand side column must be Cyan ('1') except the upper square that is Light Yellow ('2', the first prime number).
http://www.lactamme.polytechnique.fr/Mosaic/images/GILB.22.D/display.html
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. In 1878, eighty years before Gilbreath's discovery, François Proth had published the same observations.
https://en.wikipedia.org/wiki/Gilbreath's_conjecture
Proth-Gilbreath Conjecture
Beat the Andrew Odlyzko Record G(Pi(1.1x10^14^))=635 (1993)
G(Pi(10^14^))=693 on october 5 2025 at 20:59:18
G(Pi(1.145x10^14^))=744 on 10/27/2025 at 12:10:26
1-Definition:
This conjecture was stated in 1958 by Norman L. Gilbreath but published earlier in 1878 by François Proth. It is related to the prime numbers and to the sequences generated by taking the absolute value of the difference between each prime number and its successor and then repeating this process ad infinitum:
The conjecture states that the first value of each line is 1 (except the first one where it is a 2 -the only even prime number-) and was studied by Andrew Odlyzko in 1993. He did check it for all prime numbers less than 10^13^.
On sunday 10/05/2025 20:45 (Paris time, France) I did succeed to check it up to 10^14^ and on tuesday 10/07/2025 02:25 pm (East Time), Simon Plouffe (Canada) did the same. Moreover he did confirm the maximal value (693) of the G(Pi(x)) function with x ∈ [2,10^14^] that was anticipated on 09/25/2025.
2-The Theory:
Obviously one cannot check the Proth-Gilbreath Conjecture for there is an infinity of prime numbers. Only a demonstration can solve this unless a counter-example is discovered, that is a line not starting with a '1' (except the first one).
Let p~n~ be the prime numbers:
p~1~ = 2
p~2~ = 3
p~3~ = 5
etc...
Let's define the suite d~k~(n):
d~0~(n) = pn for all n such as n > 0
d~k~(n) = |d~k-1~(n) - d~k-1~(n+1)| for all k such as k > 0 and for all n such as n > 0
Then one must check that:
d~k~(1) = 1 for all k such as k > 0
Due to the finite limits of computers, it is impossible to exhaustively check this property. Fortunately Andrew Odlyzko noticed that if for a certain N there exists K such that:
d~K~(1) = 1
d~K~(n) ∈ {0,2} for all n such as 0 < n < N+1
then:
d~k~(1) = 1 for all k such as K-1 < k < N+K
Let's call G(N) the smallest k (if it exists) such that:
d~j~(1) = 1, 0 < j < k+1
d~k~(n) ∈ {0,2} for all n such as 0 < n < N+1
A trivial reasoning shows that G(N) does exist for all N and that the process can be stopped as soon as there are only '0's, '1's and '2's on the current line of rank k. For example:
More at https://www.lactamme.polytechnique.fr/Mosaic/descripteurs/GilbreathConjecture.01.Ang.html
Verification and attempted proofs
Several sources write that, as well as observing the pattern of Gilbreath's conjecture, François Proth released what he believed to be a proof of the statement that was later shown to be flawed. However, Zachary Chase disputes this, writing that although Proth called the observation a "theorem", there is no evidence that he published a proof, or false proof, of it.
[...] 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.
Simon Plouffe has announced a computational verification for the primes up to 10^14^.