Proth-Gilbreath Conjecture
Beat the Andrew Odlyzko Record G(Pi(1013))=635 (1993)
G(Pi(1014))=693 on 10/05/2025 at 20:59:18
G(Pi(2.8000*1014))=788 on 11/08/2025 at 17:46:01
G(Pi(6.1500*1014))=800 on 12/13/2025 at 17:24:56
G(Pi(1015))=800 on 01/23/2026 at 18:41:08
G(Pi(1.0025*1015))=806 on 01/24/2026 at 00:51:08
G(Pi(1.2075*1015))=809 on 02/15/2026 at 02:58:42
G(Pi(1.2125*1015))=811 on 02/15/2026 at 21:50:58
Verification up to 1.5x1015
CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France
[Site Map, Help and Search [Plan du Site, Aide et Recherche]]
[The Y2K Bug [Le bug de l'an 2000]]
[Real Numbers don't exist in Computers and Floating Point Computations aren't safe. [Les Nombres Réels n'existent pas dans les Ordinateurs et les Calculs Flottants ne sont pas sûrs.]]
[Please, visit A Virtual Machine for Exploring Space-Time and Beyond, the place where you can find more than 10.000 pictures and animations between Art and Science]
(CMAP28 WWW site: this page was created on 09/28/2025 and last updated on 03/18/2026 14:26:17 -CET-)
[en français/in french]
Preliminary Remark:
The following research is the fruit of a collaboration with Jean-Paul Delahaye professor at the Université de Lille,
researcher at the Lille Cristal laboratory and well known columnist for the Pour La Science newspaper.
Contents:
1-Introduction:
A conjecture
is a mathematical statement
that is believed to be true but cannot be called a theorem as long as every
attempt at a proof has failed...
The most beautiful recent example was Andrew Wiles's
proof of Fermat's Last Theorem in 1994, which had resisted the greatest mathematicians
since the 17th century.
Many important conjectures are currently "pending".
Such is the case for Goldbach's Conjecture -GC-
and the Twin Prime Conjecture -TPC-.
For a given conjecture, as long as no correct proof has been found, it is possible to try
to disprove it by finding a counterexample, when that makes sense. This is not the case for the
TPC, since no longer finding pairs of twin primes would in no way prove that, by searching
much further, new pairs would not appear. On the other hand, for GC, discovering
an even number greater than 2 that is not the sum of two primes would obviously prove that GC is false.
Unfortunately, if such a number exists, the likelihood of it being within our reach
is practically zero (in fact, all integers are enormous, indeed unimaginable, except,
of course, the small ones we use in everyday life...).
But even if the search for a counterexample
may sometimes seem futile, it can nevertheless be useful by revealing relevant information about
a given topic and this is the case for Gilbreath's Conjecture and prime numbers...
2-Definition:
This conjecture was stated in 1958 by Norman L. Gilbreath [02] but published earlier in 1878 by François Proth [03].
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 [04]:
2 3 5 7 11 13 17 19 23 29 31 (...) <-- Prime Numbers
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
1 2 2 4 2 4 2 4 6 2 (...) <-- Prime Gaps
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
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 (...)
\ / \ / \ / \ / \ /
\ / \ / \ / \ / \ /
\ / \ / \ / \ / \ /
\ / \ / \ / \ / \ /
1 2 0 2 (...)
\ / \ / \ / \ /
\ / \ / \ / \ /
\ / \ / \ / \ /
\ / \ / \ / \ /
1 2 2 (...)
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
1 0 (...)
\ / \ /
\ / \ /
\ / \ /
\ / \ /
1 (...)
\ /
\ /
\ /
\ /
(...)
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 1013.
On sunday 10/05/2025 20:45 (Paris time, France) I did succeed to check it up to 1014
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)) [05] function with x ∈ [2,1014]
that was anticipated on 09/25/2025.
Here are some visualizations of this process including some absolute records
(first column (left): the difference signs, second column: the difference of the absolute values and third column: the differences):
x 
= 
x 
= 
x 
= 
x 
= 
x 
= 
x 
= 
G(Pi(6.1500x1014))=800 -absolute record-
x 
= 
G(Pi(1.0025x1015))=806 -absolute record-
x 
= 
G(Pi(1.2075x1015))=809 -absolute record-
x 
= 
G(Pi(1.2125x1015))=811 -absolute record-
with the following colors regarding the numbers on the third column pictures:
0 = Dark Cyan,
-1 = Dark Orange,
+1 = Light Orange,
-2 = Dark Green,
+2 = Light Green,
when all other numbers -{3,4,5,6,7,8,...}- are Grey (Dark and Light Grey respectively for the negative and positive numbers).
When the first used prime number is 2, according to the Proth-Gilbreath Conjecture, in the middle pictures, the left-hand side column must be Cyan ('1')
except the square at the very top that is Light Yellow ('2', the first prime number).
One can notice the yellow squares are defining monodimensional binary cellular automata.
Here are some more visualizations of this process with possibly relative records:
G(Pi(1.6931x1015))=969 -relative record-
G(Pi(6.0213x1027))=1935 -relative record-
3-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 pn be the prime numbers:
p1=2
p2=3
p3=5
etc...
Let's define the suite dk(n):
d0(n) = pn for all n such as n > 0
dk(n) = |dk-1(n) - dk-1(n+1)| for all k such as k > 0 and for all n such as n > 0
Then one must check that:
dk(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:
dK(1) = 1
dK(n) ∈ {0,2} for all n such as 0 < n < N+1
then:
dk(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:
dj(1) = 1, 0 < j < k+1
dk(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:
k=0 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71
≠ ≠ ≠ ≠≠ ≠≠ ≠≠ ≠≠ ≠≠ ≠≠ ≠≠ ≠≠ ≠≠ ≠≠ ≠≠ ≠≠ ≠≠ ≠≠ ≠≠ ≠≠
k=1 1 2 2 4 2 4 2 4 6 2 6 4 2 4 6 6 2 6 4
≠ ≠ ≠ ≠ ≠ ≠ ≠ ≠ ≠ ≠ ≠
k=2 1 0 2 2 2 2 2 2 4 4 2 2 2 2 0 4 4 2
≠ ≠ ≠ ≠
k=3 1 2 0 0 0 0 0 2 0 2 0 0 0 2 4 0 2
≠
k=4 1 2 0 0 0 0 2 2 2 2 0 0 2 2 4 2
≠
k=5 1 2 0 0 0 2 0 0 0 2 0 2 0 2 2
==> G=5
4-The Computation:
In 1993, Andrew Odlyzko verified this for all prime numbers less than 1013.
On Sunday, August 10, 2025, Jean-Paul Delahaye suggested to me "a fine calculation", nontrivial but seemingly feasible: the idea was to go beyond
1013 and reach, for example, 1014...
On paper, the task is simple: one merely has to follow the example above, but instead of stopping at 71, go past 1013.
It is clear that computers can neither store nor handle such a vast quantity of numbers. The idea, therefore, is to
work in successive blocks, each of a size compatible with the machines being used.
However, this subdivision cannot be done naïvely, as can be seen by cutting the previous example into two blocks:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71
1 2 2 4 2 4 2 4 6 6 4 2 4 6 6 2 6 4
1 0 2 2 2 2 2 2 2 2 2 2 0 4 4 2
0 0 0 2 4 0 2
0 0 2 2 4 2
0 2 0 2 2
Obviously, the difference 31-29 is
missing. Jean-Paul Delahaye then suggested that the blocks should not be disjoint, but should
partially overlap as follows, based on an estimated G (taken as 5 below):
2 3 5 7 11 13 17 19 23 29 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71
1 2 2 4 2 4 2 4 6 4 2 4 6 2 6 4 2 4 6 6 2 6 4
1 0 2 2 2 2 2 2 2 2 2 4 4 2 2 2 2 0 4 4 2
1 2 0 0 0 0 0 0 0 2 0 2 0 0 0 2 4 0 2
1 2 0 0 0 0 0 2 2 2 2 0 0 2 2 4 2
1 2 0 0 0 2 0 0 0 2 0 2 0 2 2
that is:
2 3 5 7 11 13 17 19 23 29
13 17 19 23 29 31 37 41 43 47 53 59 61 67 71
== == == == ==
1 2 2 4 2 4 2 4 6
4 2 4 6 2 6 4 2 4 6 6 2 6 4
= = = =
1 0 2 2 2 2 2 2
2 2 2 4 4 2 2 2 2 0 4 4 2
= = =
1 2 0 0 0 0 0
0 0 2 0 2 0 0 0 2 4 0 2
= =
1 2 0 0 0 0
0 2 2 2 2 0 0 2 2 4 2
=
1 2 0 0 0
2 0 0 0 2 0 2 0 2 2
One might note that this overlap of 5 numbers significantly increases
the number of computations. In fact, this is true in this example, where each block
contains only 10 numbers. But the calculations that will be carried out to break the 1993 record will
use much larger blocks (generally of size 107), and even if the overlap used is
larger (typically around 1000), the number of additional computations due to the overlaps remains
relatively very small.
As for the choice of overlap size, it was guided by the results published in 1993 by Andrew Odlyzko:
x Pi(x) G(Pi(x))
102 25 5
103 168 15
104 1229 35
105 9592 65
106 78498 95
107 664579 135
108 5761455 175
109 50847534 248
1010 455052511 329
1011 4118054813 417
1012 37607912018 481
1013 346065536839 635
where 'x' is the last integer tested (the last prime number being strictly less
than x), 'Pi(x)' is the number of prime numbers in [1,x],
and finally 'G(Pi(x))' gives us the minimum overlap size.
The Hardware:
Six computers are used [06]: on the one hand, five computing servers ("C") that perform the actual
calculations, and on the other hand, a PC called the "Supervisor"
("S") that manages the distribution of the various computations among the
servers. All these machines run under Linux and are connected to an NFS file server. The configurations
below indicate, in order, the maximal number of processors [07], the memory capacity
in GB [08], and finally a performance index [09]:
- C1: {40,263,1}
- C2: {12,198,1.3875}
- C3: {12,148,1.48375}
- C4: {16,131,1.52857}
- C5: {36,528,1.6411}
The programs:
A set of programs designed to run on Linux computers has been written:
- A base program P1 in C, which is executed in parallel on all the active processors of the computing servers and receives the following arguments:
- The first integer PNE to be tested (not necessarily prime).
- The last integer DNE to be tested (not necessarily prime).
- The number of blocks NB defining the division of the interval [PNE,DNE].
- The number of prime numbers NPP defining the overlap between blocks.
- Several additional arguments, mainly used to monitor the progress of the computation.
This program defines the key function NextPrime(n), which returns the first prime number strictly greater than n,
using deterministic Miller-Rabin primality tests.
A main loop generates the NB blocks of the interval [PNE, DNE], incorporating the overlaps as needed. Then, an internal
loop successively scans each of these NB blocks to compute the differences defining the process.
In principle, this loop is repeated NNP times, but in practice, an optimization
interrupts it as soon as only '0's, '1's, or '2's remain.
At a given time, all occurences of P1 on the different servers Ci will be executed
with the lowest priority [10], which, combined with the very low memory "consumption"
[08], minimizes the inconvenience caused to other potential users.
- A script P2 in C Shell, executed on S, which evenly distributes the computations to be performed one block per
processor and waits until all the blocks have been processed.
This corresponds to a static allocation of blocks to processors.
However, since not all blocks require the same computation time [11],
it is clear that a dynamic allocation would have been preferable. To achieve this, it would
have sufficed to create smaller but more numerous blocks and place them in a queue. Then, whenever
any processor became free, it would fetch the next block from the head of the queue, continuing until the queue was empty.
However, since there are several different computing servers, this queue would have had to be an NFS-shared file.
Unfortunately, even when an NFS server is
properly configured, exclusive access during critical file update phases is not 100% reliable.
Experiments confirmed this, forcing the implementation of static allocation [12]...
- A script P3 in C Shell, executed on S, which divides the full computation into sub-tasks submitted to P2,
each requiring about five hours of processing.
This allows the entire computation to be interrupted (either intentionally or due to hardware failures)
and resumed later, losing only a few hours of work without losing useful information.
P3 also allows the launch of the different P2s to be spread out over time: this is why,
for the 10/05/2025 record, the calculations only took place during week-ends and weekday nights.
5-The Results:
The first computations started on saturday 09/20/2025 at 11:14:28 and then were suspended a few days due to the unavailability of the computing servers.
The final computation was completed [13] on Sunday, 10/05/2025, at 20:59:18 p.m. (Paris time,
France), as I reported in an email to Jean-Paul Delahaye at 10:36:20 p.m., informing
him of the end of the final stage (which concerned the prime numbers from 75,000,000,000,000
to 100,000,000,000,000). This confirmed an email I had sent him earlier,
on 09/25/2025, at 7:17:47 a.m.:
Bonne nouvelle: G(Pi(2x1013))=693
which indicated that G(Pi(1014)) would be greater than or equal to 693.
Therefore, on 10/05/2025, at 20:59:18 p.m., I was able to state:
G(Pi(1014))=693
The record held by Andrew Odlyzko since 1993 had been broken!
Simon Plouffe [14] announced on tuesday 10/07/2025, at 02:25 p.m. (Eastern Time)
that he had also found G(Pi(1014))=693, more than 24 hours after me...
It is very important to note that Simon Plouffe and I did not use the same methods or the same programs [15].
The fact that we both obtained the same result independently guarantees the accuracy of the value 693.
Here are some data regarding the processor usage during the first record computation:
The computation currently continues beyond 1x1014 and starting on 11/10/2025 is inside [3x1014,10x1014].
New records appear:
6-The Successive Absolute Records [01]:
G(Pi(x)) = G(Pi(1014))=693 x ∈ [19563862928347,19595015607259] -10/05/2025, 20:59:18-
G(Pi(x)) = G(Pi(1.1000x1014))=701 x ∈ [108623442367669,108652648007513] -10/24/2025, 16:45:28-
G(Pi(x)) = G(Pi(1.1450x1014))=744 x ∈ [144875389408103,144906542089667] -10/27/2025, 12:10:26-
G(Pi(x)) = G(Pi(2.2000x1014))=773 x ∈ [218200934579443,218228193179959] -11/03/2025, 01:39:05-
G(Pi(x)) = G(Pi(2.8000x1014))=788 x ∈ [277889408099683,277908878538719] -11/08/2025, 17:46:01-
G(Pi(x)) = G(Pi(6.1500x1014))=800 x ∈ [613228193146423,613255451754013] -12/13/2025, 17:24:56-
G(Pi(x)) = G(Pi(1015))=800 x ∈ [613228193146423,613255451754013] -01/23/2026, 18:41:08-
G(Pi(x)) = G(Pi(1.0025x1015))=806 x ∈ [1002157201513887,1002157320872243] -01/24/2026, 00:51:08-
G(Pi(x)) = G(Pi(1.2075x1015))=809 x ∈ [1206596573208751,1206627725898989] -02/15/2026, 02:58:42-
G(Pi(x)) = G(Pi(1.2125x1015))=811 x ∈ [1212869937694721,1212889408142341] -02/15/2026, 21:50:58-
On 03/18/2026,
15x4x10x96
= 57600
values of G(Pi(x)) have already been obtained.
Here they are presented, grouped into sets of
8x96
= 768
and averaged:
N=1 Moyenne768(G)=470 |----------------------------------------------*
N=2 Moyenne768(G)=494 |------------------------------------------------*
N=3 Moyenne768(G)=503 |-------------------------------------------------*
N=4 Moyenne768(G)=509 |-------------------------------------------------*
N=5 Moyenne768(G)=516 |--------------------------------------------------*
N=6 Moyenne768(G)=514 |--------------------------------------------------*
N=7 Moyenne768(G)=516 |--------------------------------------------------*
N=8 Moyenne768(G)=520 |---------------------------------------------------*
N=9 Moyenne768(G)=522 |---------------------------------------------------*
N=10 Moyenne768(G)=523 |---------------------------------------------------*
N=11 Moyenne768(G)=528 |---------------------------------------------------*
N=12 Moyenne768(G)=529 |---------------------------------------------------*
N=13 Moyenne768(G)=527 |---------------------------------------------------*
N=14 Moyenne768(G)=528 |---------------------------------------------------*
N=15 Moyenne768(G)=529 |---------------------------------------------------*
N=16 Moyenne768(G)=532 |----------------------------------------------------*
N=17 Moyenne768(G)=530 |----------------------------------------------------*
N=18 Moyenne768(G)=532 |----------------------------------------------------*
N=19 Moyenne768(G)=534 |----------------------------------------------------*
N=20 Moyenne768(G)=534 |----------------------------------------------------*
N=21 Moyenne768(G)=534 |----------------------------------------------------*
N=22 Moyenne768(G)=540 |-----------------------------------------------------*
N=23 Moyenne768(G)=537 |----------------------------------------------------*
N=24 Moyenne768(G)=538 |----------------------------------------------------*
N=25 Moyenne768(G)=537 |----------------------------------------------------*
N=26 Moyenne768(G)=538 |----------------------------------------------------*
N=27 Moyenne768(G)=537 |----------------------------------------------------*
N=28 Moyenne768(G)=539 |----------------------------------------------------*
N=29 Moyenne768(G)=541 |-----------------------------------------------------*
N=30 Moyenne768(G)=541 |-----------------------------------------------------*
N=31 Moyenne768(G)=543 |-----------------------------------------------------*
N=32 Moyenne768(G)=543 |-----------------------------------------------------*
N=33 Moyenne768(G)=542 |-----------------------------------------------------*
N=34 Moyenne768(G)=543 |-----------------------------------------------------*
N=35 Moyenne768(G)=541 |-----------------------------------------------------*
N=36 Moyenne768(G)=544 |-----------------------------------------------------*
N=37 Moyenne768(G)=543 |-----------------------------------------------------*
N=38 Moyenne768(G)=548 |-----------------------------------------------------*
N=39 Moyenne768(G)=544 |-----------------------------------------------------*
N=40 Moyenne768(G)=547 |-----------------------------------------------------*
N=41 Moyenne768(G)=546 |-----------------------------------------------------*
N=42 Moyenne768(G)=548 |-----------------------------------------------------*
N=43 Moyenne768(G)=546 |-----------------------------------------------------*
N=44 Moyenne768(G)=548 |-----------------------------------------------------*
N=45 Moyenne768(G)=548 |-----------------------------------------------------*
N=46 Moyenne768(G)=549 |-----------------------------------------------------*
N=47 Moyenne768(G)=547 |-----------------------------------------------------*
N=48 Moyenne768(G)=549 |-----------------------------------------------------*
N=49 Moyenne768(G)=544 |-----------------------------------------------------*
N=50 Moyenne768(G)=548 |-----------------------------------------------------*
N=51 Moyenne768(G)=547 |-----------------------------------------------------*
N=52 Moyenne768(G)=548 |-----------------------------------------------------*
N=53 Moyenne768(G)=549 |-----------------------------------------------------*
N=54 Moyenne768(G)=550 |------------------------------------------------------*
N=55 Moyenne768(G)=552 |------------------------------------------------------*
N=56 Moyenne768(G)=554 |------------------------------------------------------*
N=57 Moyenne768(G)=551 |------------------------------------------------------*
N=58 Moyenne768(G)=552 |------------------------------------------------------*
N=59 Moyenne768(G)=551 |------------------------------------------------------*
N=60 Moyenne768(G)=551 |------------------------------------------------------*
N=61 Moyenne768(G)=552 |------------------------------------------------------*
N=62 Moyenne768(G)=553 |------------------------------------------------------*
N=63 Moyenne768(G)=550 |------------------------------------------------------*
N=64 Moyenne768(G)=553 |------------------------------------------------------*
N=65 Moyenne768(G)=554 |------------------------------------------------------*
N=66 Moyenne768(G)=555 |------------------------------------------------------*
N=67 Moyenne768(G)=554 |------------------------------------------------------*
N=68 Moyenne768(G)=553 |------------------------------------------------------*
N=69 Moyenne768(G)=555 |------------------------------------------------------*
N=70 Moyenne768(G)=553 |------------------------------------------------------*
N=71 Moyenne768(G)=554 |------------------------------------------------------*
N=72 Moyenne768(G)=556 |------------------------------------------------------*
N=73 Moyenne768(G)=557 |------------------------------------------------------*
N=74 Moyenne768(G)=554 |------------------------------------------------------*
N=75 Moyenne768(G)=552 |------------------------------------------------------*
Obviously, the overall trend is towards groth, but very very slowy...
The computation up to 1.5x1015 was interrupted at that date, whereas it could have
continued for billions and billions of years (and much, much longer after the transition from 64 to 128 bits...)!
A small remark:
1015 (or more...) may seem like an enormous value, but in fact it is not. Indeed,
all integers are unimaginable, inaccessible,...
except for the first ones, obviously. Yet the same is true of the prime numbers,
since their set is infinite. Thus, if there exists a counter example to the Proth-Gilbreath
conjecture, the probability that it is accessible is most certainly almost zero.
Here is an update of the array published in 1993 by Andrew Odlyzko:
x : Pi(x) : G(Pi(x)) : Date :
1013 346065536839 635 Andrew Odlyzko (1993).
1014 3204941750857 693 Jean-François Colonna (10/05/2025), Simon Plouffe (10/07/2025).
1015 29844570423226 800 Jean-François Colonna (01/23/2026).
7-The Successive Relative Records [01]:
G(Pi(x)) = 1347 x ∈ [5733241593241096731,5733241593241296731] -02/27/2026, 12:33:21-
G(Pi(x)) = 1559 x ∈ [20733746510561342863,20733746510561542863] -03/01/2026, 17:51:08-
G(Pi(x)) = 1935 x ∈ [6021312001028259683626468703,6021312001028259683626668637] -03/06/2026, 18:52:00-
8-The Prime Gaps:
Simultaneously the primes gaps were computed and possibly associated with the absolute or relative records obtained [01]:
Prime Numbers : Prime Gaps : G(Pi(x)) :
[64-bit Computations]
2 1
3 2
7 4
23 6
89 8
113 14
523 18
887 20
1129 22
1327 34
9551 36
15683 44
19609 52
31397 72
155921 86
360653 96
370261 112
492113 114
1349533 118
1357201 132
2010733 148
4652353 154
17051707 180
20831323 210
47326693 220
122164747 222
189695659 234
191912783 248
387096133 250
436273009 282
1294268491 288
1453168141 292
2300942549 320
3842610773 336
4302407359 354
10726904659 382
20678048297 384
22367084959 394
25056082087 456
42652618343 464
127976334671 468
182226896239 474
241160624143 486
297501075799 490
303371455241 500
304599508537 514
416608695821 516
461690510011 532
614487453523 534
738832927927 540
1346294310749 582
1408695493609 588
1968188556461 602
2614941710599 652
7177162611713 674
13829048559701 716
19581334192423 766 693 (Absolute Record)
42842283925351 778
90874329411493 804
171231342420521 806
218209405436543 906 773 (Absolute Record)
1189459969825483 916
1686994940955803 [16] 924 729
1693182318746371 1132 969
43841547845541059 1184 1199
55350776431903243 1198 439
80873624627234849 1220 1073
203986478517455989 1224 969
218034721194214273 1248 907
305405826521087869 1272 1059
352521223451364323 1328 1245
401429925999153707 1356 1057
418032645936712127 1370 1113
804212830686677669 1442 1083
1425172824437699411 1476 1218
5733241593241196731 1488 1347 (Relative Record)
6787988999657777797 1510 1096
[128-bit Computations]
15570628755536096243 1526 329
17678654157568189057 1530 1316
18361375334787046697 1550 1305
18470057946260698231 1552 1325
18571673432051830099 1572 1200
20733746510561442863 1676 1559 (Relative Record)
68068810283234182907 1724 1515
88409025774659694609269 1628 1320
88409026124861029148819 1574 1375
90111023769130809399029 1580 1158
91008005685955879916401 1612 1322
92008005233975799174049 1584 1521
93310024192891977362587 1624 1136
153125481414651411510001 1588 1357
157169790357596057379929 1668 1327
227125476252860770095113 1604 1417
275125480759770276082019 1644 1503
348125468109010265299271 1602 1263
444888024000161880076721 1578 1179
461003025023148162667033 1650 1097
475135024904107611376237 1750 495
506169788449647021511111 1638 1237
507139024133574480779581 1590 1243
511125438041415299421601 1572 1346
519125479346441609466023 1596 1412
601559025236668271908379 1620 439
645600025854624019250411 1598 1295
654774025979872326628579 1570 1176
671442875163990116829497 1632 1194
708664733765282327176751 1710 1543
808129786082486355062717 1586 1323
817125469610972857731061 1692 1437
822125462857195122741779 1634 1281
829125472095488458309583 1626 1517
908600025159019423417933 1654 1413
929125470064749912335467 1600 1429
1409619025494896363461087 1680 1355
1689619025236598760976829 1610 1320
1863007027058448917031433 1726 1525
2417837025702997028843443 1636 1308
2668123000582492119464617 1656 1333
4417837025271189172082513 1608 1312
4591958808448499496305287 1780 1257
4601311025550610228138711 1690 1579
4771561025190978473357407 1582 1317
4871005025825741055913559 1694 1501
5013267025248268009777159 1672 1226
5111827025410880328372583 1606 1379
5701963804766267176483453 1594 1072
9242091025170776894993897 1766 1427
10148375028695776627340501 1640 965
10234026002878309225086799 1648 1370
10234026003107509182675341 1676 1173
10503126003077789930911879 1630 1369
17853322005951711516168487 1686 1411
20383026002229098626547639 1592 1387
30011026001827477703636173 1740 1497
37039655026170596199293657 1752 391
37039655026277896461956917 1746 1439
38283026000105876456447549 1658 1307
39753026000841430736202739 1660 1457
40503126001716558768993799 1662 1313
42461577000200688572056091 1728 1540
50011026000605167554818833 1614 1430
50011026000985432236994141 1576 1001
50011026001254085912788547 1642 1232
50148375026035070947338349 1678 1433
50234026000970339549409247 1666 1424
50503126001280209002157267 1616 1499
50754026000858414432532293 1734 1456
55723122040086608203258649 1698 1435
69361551050100334908440329 1702 989
72753026003121484051877369 1754 1369
73283026003973067104501339 1674 1469
79139658026044243740847151 1646 1389
80184269026465047227041177 1722 1575
83753026001934316187838497 1682 1375
84199650026160877110518461 1732 1402
84523126003293348705947939 1622 1581
86087256026428977676556479 1830 1470
86283026003161194576775057 1762 1301
88148378027905948598974153 1618 957
90523126001136840586142237 1652 1341
90523126002000439101137111 1776 1617
90523126002091526467218463 1744 998
90523126002117844475997013 1704 1479
91084263026046832262353511 1808 1615
91283026002272823778115587 1854 1580
94283026000460650817643547 1806 1291
96523126000774442796493001 1718 1055
110449654697555438112561319 1684 1379
124127829027210048144492593 1736 1160
161023337027912452750278023 1664 1445
161023337027994323152086013 1936 1642
161023337028310365631639753 1696 721
161023337028376633976274427 1774 1347
167021551028816331494487719 1814 1103
169206313035910497586224637 1716 1427
169206313037524085550011021 1760 1111
172233338035577973225937699 1708 931
172233338035661427996026473 1860 1551
183245698035574395968790427 1872 1010
183245698038521377016396923 1804 1330
185265722028138732845783509 1812 1541
191506729036081185032535079 1720 1140
191506729036347786101707879 1824 1373
191506729036672524811449829 1758 1671
191506729038361825489489313 1844 1197
197021551027975628706026789 1688 1562
198245298028246350476778643 1764 1365
198245298028294569694862807 1724 1574
211506729033894836038769081 1800 1559
227021551027737529987065469 1810 1485
253022127027824187693016727 1790 1581
261023337027413737623820943 1706 1273
302233338032699490171225683 1730 1501
313245698033230889150530529 1700 1428
314206313032559556437705819 1778 1598
333245298028562639400861353 1670 1213
361023337029228675738125911 1818 1053
361023337029651191344064387 1770 1185
363253313027185154998755239 1712 1527
414127829027107949907596749 1768 1311
508253313027343345210322131 1786 1422
512233338030056680994432863 1840 1027
517021551027178550547527491 1798 1573
531506722030087317322633423 1756 1626
531506722032495269637700891 1870 1353
531506722032894991035261461 1788 1599
612233338029038274818850137 1866 1587
612233338029577635338157403 1834 1536
622337118707239085237867807 1850 1146
644853313030609631743421827 1792 1563
663853313028090003058817561 1796 1585
685245298027055418345996361 1990 1531
931506729027818356495510979 1772 290
931506729027974407041540707 1782 1281
1985698761742223986651814371 1738 1129
2518970080811424228048297851 1848 1721
2681804411030756891181329527 1816 1682
2681804411030796838102390267 1794 1591
2844230000030892453360363713 1920 1390
3001549619028368687092343141 1842 1384
3001549619029361061238343309 1934 1636
3039248000030181434897238311 1916 1839
3044230000030128405583745033 1944 1809
3281312000028041064344397077 1742 1195
3521777371029774497617670861 1902 1606
3681804411029893987474470217 1714 1427
3717737905029238085958241901 1748 1429
3733289118707267491829280101 1890 1589
3883778117029429378726839157 1852 1625
3883778117029553360606784223 1914 1661
4101238311033979111838291509 1858 880
4392804445028608016009081833 1918 1417
4521777371028573501293031019 1954 1669
4521777371028957272039263763 1886 1777
4603737621033901941540708091 1822 1628
4611238371029716324077383681 1922 1623
4883778117028144439778057967 1900 1433
5020547613028019048897910089 1784 1587
5570675040016229618538921497 1976 1465
5603737621032667844132289397 1894 1391
5739248000028792850873302491 1820 1289
5844230000028765302725127593 1884 1129
5851230000021967795781669357 1856 1186
5972248000023708695939463647 1826 1213
6021312001028259683626568699 1994 1935 (Relative Record)
6101238311031688672705309009 1924 1401
6101239311029812830316130021 1802 559
6101239311030479007754228103 1946 1727
6139248000028643882072689133 1974 1372
6670605040015867559498340553 1960 1689
7051230000020674054592576303 1838 1343
7500230000000254312587886349 1832 1517
7500230000004410741095419811 1836 1317
7500230000005019060037933673 1888 1482
7500230000005824418875087691 1846 1600
7500230000011523034496281371 1982 1737
8012239000018115133439311463 1938 1769
8051230000019922137852468729 1980 1667
8101238311029153282478027583 1910 1749
8104778551030016778369544613 1868 1275
8511230000017373935165665319 1882 1621
8761007116028125697454115487 1892 1312
15251000000439240915164391943 1978 1597
15600350800200231835802693153 1898 1291
15600350800200591840837572687 1950 1777
16107428778765206909298916249 1948 1547
16107428778765313805063526049 1942 1435
16107428778766062769433544071 1862 1752
16817006514738017659250207459 1968 1595
16967428778771122673539497533 1908 1307
17361011751029174933335986203 1964 1580
17817006514740891827868262213 1896 1233
21119471029029007664840376149 1878 1743
21124471029101398728988856797 1926 1504
29045119029001077581773468231 1932 1412
29045119029001231341899676133 1828 1501
78965098027967495955787825381 1972 1240
83723413734029484073263062011 1962 1561
93010225055029330341377130337 1912 1252
93777234437032666433522975959 1930 1215
93777234437033872048058808383 1928 1216
93825139554000834758506871159 1880 901
94381705258295181151295669737 1876 1827
98731233432030301788380501897 1874 1345
484458149616032368701456021079 1864 1467
631337750259693153112235938121 1958 1656
801451149611030245636120895591 1992 1613
801451149611030624254364873641 1966 1467
892340838549031853833349514503 1940 1620
917123687913032561163805565423 1988 1275
927619616117030224870646906207 1904 1365
1611281318552031378980232272063 1998 1543
2304607319542032497749866213061 1956 1633
3809205709418032358337519745171 1986 1848
5887250233904031284472398660693 1970 1429
20282409603651671079340215344953 1984 1087
45154861419265032079255464330641 1952 1654
65013315500001000157495421077531 1906 1191
85982514713000000005643994785767 1996 1801
Copyright © Jean-François COLONNA, 2025-2026.
Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2025-2026.
