New conjectures
In the first part of this long post, I propose some conjectures.

Following the posts #384, #387 and #392, I add here some conjectures that I could formulate by observing the tables produced by the analysis algorithms of Edwin Hall and mine.

I tried to classify these conjectures, but the classification is just intuitive and not based on real arguments.

- The conjectures with a star (*): I think they are true and could probably be demonstrated by mathematicians working on number theory.

Some of them are certainly already known or are even theorems. Sorry if some of these conjectures are trivial, I'm not necessarily able to realize it!

- The conjectures with two stars (**): I think they are true. If they are indeed true, they must be very difficult to prove.

But I may be wrong about some of them and further calculations should make it possible to invalidate them.

- The conjectures without a star: they are more observations than conjectures. My intuition tells me that the continuation of the calculations would show that they are false. But you never know...

- Bold conjecture is either very beautiful or spectacular!

I have been unable to produce a general conjecture. The conjectures I propose here only concern occurrences of a prime number in a base.

I cannot state all the conjectures, there are far too many. I have tried to select a representative sample.

After the statement of the conjectures, in a second part, I propose an explanation which shows how I proceed from the observation of the tables, to state these conjectures.

Finally, in the third part, I make some general remarks and ask some questions.

__I) Statement of new conjectures :__
In all the statements below, k is an integer.

__Some new conjectures for base 2:__
Note: Several of these conjectures motivated my request to Edwin Hall to push the calculations further for some exponents i=36*k, i=60*k, i=70*k, i=72*k, i=90*k.

Conjecture (1)* :

The prime number 3 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 2^(2*k).

Conjecture (2)** :

The prime number 3 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 2^(4*k).

Conjecture (3) :

The prime number 3 appears in the decomposition of the terms of indexes 1 through 7 of all sequences that begin with the integers 2^(36*k).

Conjecture (4) :

The prime number 3 appears in the decomposition of the terms of indexes 1 through 18 of all sequences that begin with the integers 2^(126*k).

Conjecture (5)* :

The prime number 5 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 2^(4*k).

Conjecture (6)** :

The prime number 5 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 2^(28*k), 2^(44*k), 2^(76*k), 2^(92*k), 2^(116*k).

Conjecture (7) :

The prime number 5 appears in the decomposition of the terms of indexes 1, 2, 3, 4 of all sequences that begin with the integers 2^(36*k).

Conjecture (8) :

The prime number 5 appears in the decomposition of the terms of indexes 1, 2, 3, 4 of all sequences that begin with the integers 2^(132*k).

Conjecture (9)* :

The prime number 7 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 2^(3*k).

Conjecture (10)** :

The prime number 7 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 2^(12*k).

Conjecture (11) :

The prime number 7 appears in the decomposition of the terms of indexes 1, 2, 3, 4 of all sequences that begin with the integers 2^(60*k).

Conjecture (12)* :

The prime number 11 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 2^(10*k).

Conjecture (13)** :

The prime number 11 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 2^(120*k), 2^(130*k).

Conjecture (14) :

The prime number 11 appears in the decomposition of the terms of indexes 1, 2, 3 of all sequences that begin with the integers 2^(70*k).

Conjecture (15) :

The prime number 13 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 2^(12*k).

Conjecture (16)** :

The prime number 13 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 2^(60*k).

Conjecture (17)* :

The prime number 17 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 2^(8*k).

Conjecture (18)** :

The prime number 17 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 2^(144*k).

Conjecture (19) :

The prime number 19 appears in the decomposition of the terms of indexes 1, 2, 3 of all sequences that begin with the integers 2^(72*k).

~~Conjecture (20) :~~

The prime number 31 appears in the decomposition of the terms of indexes 1, 2, 3 of all sequences that begin with the integers 2^(90*k).
Conjecture invalidated by Edwin Hall's calculations.

**Conjecture (21)** :**

The prime number 79 appears in the decomposition of the terms of indexes 1, 2, 3 of all sequences that begin with the integers 2^(156*k).
Conjecture (22)** :

The prime number 2089 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 2^(87*k).

Conjecture (23)** :

The prime number 4051 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 2^(100*k).

Conjecture (24) :

The prime number 15121 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 2^(540*k).

__Some new conjectures for base 3:__
Conjecture (25)*:

The prime number 5 appears in the decomposition of the terms of index 1 of all sequences starting with the integers 3^(4*k).

Conjecture (26)** :

The prime number 5 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 3^(4+8*k).

Conjecture (27)* :

The prime number 7 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 3^(6*k).

Conjecture (28)* :

The prime number 7 appears in the decomposition of the terms of many consecutive indexes of all sequences that begin with the integers 3^(6+12*k).

Here is the observation that led to this conjecture:

Code:

prime 7 in sequence 3^6 at index i for i from 1 to 5
prime 7 in sequence 3^18 at index i for i from 1 to 50
prime 7 in sequence 3^30 at index i for i from 1 to 25
prime 7 in sequence 3^42 at index i for i from 1 to 86
prime 7 in sequence 3^54 at index i for i from 1 to 179
prime 7 in sequence 3^66 at index i for i from 1 to 39
prime 7 in sequence 3^78 at index i for i from 1 to 124
prime 7 in sequence 3^90 at index i for i from 1 to 171
prime 7 in sequence 3^102 at index i for i from 1 to 72
prime 7 in sequence 3^114 at index i for i from 1 to 45
prime 7 in sequence 3^126 at index i for i from 1 to 60
prime 7 in sequence 3^138 at index i for i from 1 to 230
prime 7 in sequence 3^150 at index i for i from 1 to 148
prime 7 in sequence 3^162 at index i for i from 1 to 228
prime 7 in sequence 3^174 at index i for i from 1 to 219
prime 7 in sequence 3^186 at index i for i from 1 to 9
prime 7 in sequence 3^198 at index i for i from 1 to 105
prime 7 in sequence 3^210 at index i for i from 1 to 194
prime 7 in sequence 3^222 at index i for i from 1 to 98
prime 7 in sequence 3^234 at index i for i from 1 to 87
prime 7 in sequence 3^246 at index i for i from 1 to 38

But on reflection, this conjecture is not extraordinary.

7 is a prime number which is in the dcomposition of the 2^2*7 driver.

It is therefore normal that it persists in so many consecutive terms.

On the other hand, it should be shown here that s(3^(6+12*k)) has the driver 2^2*7 as a factor.

Conjecture (29)* :

The prime number 11 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 3^(5*k).

Conjecture (30)* :

The prime number 13 appears in the decomposition of index 1 terms in all sequences that begin with the integers 3^(3*k).

Conjecture (31)** :

The prime number 13 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 3^(51*k).

Conjecture (32)* :

The prime number 17 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 3^(16*k).

Conjecture (33)** :

The prime number 17 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 3^(48*k).

Conjecture (34) (* if only index 1) :

The prime number 19 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 3^(18*k).

Conjecture (35) (* if only index 1 and 2):

The prime number 19 appears in the decomposition of the terms of indexes 1, 2, 3 of all sequences that begin with the integers 3^(36*k).

Conjecture (36)* :

The prime number 23 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 3^(11*k).

Conjecture (37)** :

The prime number 31 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 3^(30*k).

Conjecture (38)* :

The prime number 37 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 3^(18*k).

Conjecture (39)** :

The prime number 37 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 3^(36*k).

**Conjecture (40)** :**

The prime number 79 appears in the decomposition of the terms of indexes 1, 2, 3 of all sequences that begin with the integers 3^(78*k).

Conjecture (41)** :

The prime number 547 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 3^(14*k).

Conjecture (42)**, already known conjecture, see previous posts :

The prime number 398581 appears in the decomposition of the terms of indexes 1, 2, 3 of all sequences that begin with the integers 3^(26*k).

__Some new conjectures for base 5:__
Conjecture (43):

The prime number 3 appears in the decomposition of the terms of index 1 of all sequences starting with the integers 5^(2*k).

**Conjecture (44) :**

The prime number 3 appears in the decomposition of many consecutive indexes of all sequences that begin with the integers 5^(2+4*k).

For example, 3 appears in the decomposition of the terms in indexes 1 through 786 of the sequence that begins with 5^58.
Conjecture (45)* :

The first number 5 never appears in the decomposition of the terms at index 1 of all sequences beginning with the integers 5^(k).

Conjecture (46)* :

The prime number 7 appears in the decomposition of terms at index 1 of all sequences that begin with the integers 5^(6*k).

Conjecture (47)** :

The prime number 7 appears in the decomposition of index 1 and index 2 terms of all sequences that begin with the integers 5^(12*k).

Conjecture (48)* :

The prime number 11 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 5^(5*k).

Conjecture (49)** :

The prime number 11 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 5^(35*k).

Conjecture (50)** :

The prime number 11 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 5^(40*k).

Conjecture (51)** :

The prime number 11 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 5^(65*k).

Conjecture (52)* :

The prime number 13 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 5^(4*k).

Conjecture (53)* :

The prime number 17 appears in the decomposition of index 1 terms in all sequences that begin with the integers 5^(16*k).

Conjecture (54)* :

The prime number 19 appears in the decomposition of index 1 terms in all sequences that begin with the integers 5^(9*k).

Conjecture (55)** :

The prime number 19 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 5^(18*k).

Conjecture (56)* :

The prime number 31 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 5^(3*k).

Conjecture (57)** :

The prime number 31 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 5^(30*k).

Conjecture (58) :

The prime number 31 appears in the decomposition of the terms of many consecutive indexes of all sequences that begin with the integers 5^(48+96*k).

Here is the observation that led to this conjecture:

Code:

prime 31 in sequence 5^48 at index i for i from 1 to 447
prime 31 in sequence 5^144 at index i for i from 1 to 32

The same remark can be made here as for the conjecture (28).

And it should be shown here that s(5^(48+96*k)) has the driver 2^4*31 as a factor.

Conjecture (59)* :

The prime number 71 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 5^(5*k).

Conjecture (60)** :

The prime number 71 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 5^(45*k).

Conjecture (61)* :

The prime number 521 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 5^(10*k).

Conjecture (62)** :

The prime number 521 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 5^(50*k).

__Some new conjectures for base 6:__
Conjecture (63):

The prime number 3 appears in the decomposition of the terms of index 1 of all sequences that start with the integers 6^(1+2*k).

Conjecture (64)* :

The prime number 5 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 6^(2*k).

Conjecture (65)* :

The prime number 7 appears in the decomposition of index 1 terms of all sequences that begin with the integers 6^(6*k).

Conjecture (66)* :

The prime number 11 appears in the decomposition of index 1 terms of all sequences that begin with the integers 6^(10*k) and 6^(2+10*k).

Conjecture (67)* :

The prime number 13 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 6^(12*k).

Conjecture (68)* :

The prime number 19 appears in the decomposition of index 1 terms of all sequences that begin with the integers 6^(18*k) and 6^(10 + 18*k).

Conjecture (69)* :

The prime number 23 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 6^(11*k).

Conjecture (70)* :

The prime number 29 appears in the decomposition of index 1 terms in all sequences that begin with the integers 6^(28*k).

Conjecture (71)* :

The prime number 31 appears in the decomposition of index 1 terms of all sequences that begin with the integers 6^(30*k), 6^(11+30*k) and 6^(17 + 30*k).

Conjecture (72)** :

The prime number 37 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 6^(36*k)

Conjecture (73)* :

The prime number 37 appears in the decomposition of the terms of index 1 of all sequences starting with the integers 6^(14+36*k).

Conjecture (74)* :

The prime number 59 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 6^(58*k), 6^(8+58*k), 6^(35+58*k) and 6^(53+58*k).

Conjecture (75)* :

The prime number 61 appears in the decomposition of the terms of index 1 of all sequences beginning with the integers 6^(60*k), 6^(44+60*k) and 6^(55+60*k).

Conjecture (76)* :

The prime number 71 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 6^(70*k), 6^(11+70*k), 6^(32+70*k), 6^(35+70*k), 6^(46+70*k) and 6^(67+70*k).

Conjecture (77)* :

The prime number 601 appears in the decomposition of the terms of index 1 of all sequences beginning with the integers 6^(75*k).

__Some new conjectures for Base 7:__

Conjecture (78):

The prime number 3 appears in the decomposition of the terms of index 1 of all sequences starting with the integers 7^(3*k).

Conjecture (79) :

The prime number 3 appears in the decomposition of the terms from index 1 to 10 for all sequences that begin with the integers 5^(6+12*k) and 5^(21*k).

Conjecture (80)* :

The prime number 5 appears in the decomposition of the terms of index 1 for all sequences that begin with the integers 7^(4*k).

Conjecture (81)* :

The prime number 7 appears in the decomposition of index 1 terms in all sequences that begin with the integers 7^(3*k).

Conjecture (82)* :

The prime number 7 never appears in index 1 of all sequences that begin with the integers 7^(k).

Conjecture (83)* :

The prime number 11 appears in the decomposition of the terms in index 1 of all sequences that begin with the integers 7^(10*k).

Conjecture (84)** :

The prime number 13 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 7^(12*k).

**Conjecture (85) :**

The prime number 13 appears in the decomposition of the terms of many indexes of all sequences that begin with the integers 7^(72*k).

27 consecutive indexes for 7^72 and 9 consecutive indexes for 7^144.
Conjecture (86)* :

The prime number 17 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 7^(16*k).

Conjecture (87)** :

The prime number 17 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 7^(32*k).

Conjecture (88)* :

The prime number 19 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 7^(3*k).

Conjecture (89)** :

The prime number 19 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 7^(9*k).

Conjecture (90)* :

The prime number 31 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 7^(15*k).

Conjecture (91)** :

The prime number 67 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 7^(66*k).

Conjecture (92)* :

The prime number 419 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 7^(19*k).

**Conjecture (93)** :**

The prime number 419 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 7^(38*k).

Checked up to k=23.
__Some new conjectures for base 10:__
Conjecture (94):

The prime number 3 appears in the decomposition of the terms of index 1 of all sequences starting with the integers 10^(2*k).

Conjecture (95)* :

The prime number 5 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 10^(3+4*k).

Conjecture (96)* :

The prime number 13 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 10^(12*k).

Conjecture (97)** :

The prime number 13 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 10^(2+12*k).

Conjecture (98)** :

The prime number 19 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 10^(18*k).

Conjecture (99)* :

The prime number 61 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 10^(52+60*k) and 10^(54+60*k).

Conjecture (100)** :

The prime number 61 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 10^(60*k).

__Some new conjectures for base 11 :__
Conjecture (101)*:

The prime number 7 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 11^(3*k).

Conjecture (102)** :

The prime number 7 appears in the decomposition of the terms of indexes 1, 2, 3 of all sequences that begin with the integers 11^(6*k).

Conjecture (103)* :

The prime number 11 never appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 11^(k).

**Conjecture (104)** :**

The prime number 13 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 11^(k).
Conjecture (105)* :

The prime number 19 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 11^(3*k).

Conjecture (106)** :

The prime number 19 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 11^(6*k).

**Conjecture (107)** :**

The product of prime 19*79*547 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 11^(39*k).

REMARKABLE, checked up to k=12. See prime 79 bases 2 and 3.
__Some new conjectures for base 12:__
Conjecture (108):

The prime number 3 never appears in the decomposition of the terms of index 1 of all sequences that start with the integers 12^(k).

Conjecture (109)* :

The prime number 17 appears in the decomposition of index 1 terms in all sequences that begin with the integers 12^(16*k) and 12^(6+16*k).

It is difficult to notice for base 12, other behaviors different from the bases already presented so far.

__Some new conjectures for base 13:__

Conjecture (110)* :

The prime number 3 appears in the decomposition of the terms of index 1 of all sequences that start with the integers 13^(3*k).

**Conjecture (111) :**

The prime number 3 appears in the decomposition of the terms of indexes 1 through 6 of all sequences that begin with the integers 13^(6*k).
Conjecture (112)* :

The prime number 5 appears in the decomposition of the terms in index 1 of all sequences that begin with the integers 13^(4*k).

Conjecture (113) :

The prime number 5 appears in the decomposition of the terms of many consecutive indexes of all sequences that begin with the integers 13^(8+16*k).

Conjecture (114)* :

The prime number 7 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 13^(2*k).

Conjecture (115) :

The prime number 7 appears in the decomposition of the terms of many consecutive indexes of all sequences that begin with the integers 13^(4+8*k).

Conjecture (116)* :

The prime number 13 never appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 13^(k).

Conjecture (117)* :

The prime number 19 appears in the decomposition of index 1 terms in all sequences that begin with the integers 13^(18*k).

Conjecture (118)** :

The prime number 19 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 13^(36*k).

Conjecture (119)* :

The prime number 29 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 13^(14*k).

Conjecture (120)** :

The prime number 29 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 13^(42*k).

Conjecture (121)* :

The prime number 61 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 13^(3*k).

Conjecture (122)** :

The prime number 61 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 13^(21*k).

__Some new conjectures for base 14:__

**Conjecture (123)** :**

The prime number 3 appears in the decomposition of the terms of indexes 1 to 4 of all sequences starting with the integers 14^(6*k).
Conjecture (124)* :

The prime number 5 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 14^(4*k).

Conjecture (125)** :

The prime number 5 appears in the decomposition of the terms of indexes 1 to 4 of all sequences that begin with the integers 14^(1+4*k).

It is difficult to notice for base 14, other behaviors different from the bases already presented so far...

__Some new conjectures for base 15 :__
Conjecture (126)* :

The prime number 7 appears in the decomposition of the terms of index 1 of all sequences which start with the integers 15^(2*k), except for the 15^(8+12*k).

__Some new conjectures for base 17 :__
Conjecture (127)** :

The prime number 3 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 17^(2*k).

Conjecture (128)** :

The prime number 5 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 17^(4*k).

Conjecture (129)** :

The prime number 7 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 17^(6*k).

Conjecture (130)** :

The prime number 19 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 17^(9*k).

Conjecture (131)** :

The prime number 19 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 17^(36*k).

Conjecture (132)** :

The prime number 229 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 17^(19*k).

Conjecture (133)** :

The prime number 229 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 17^(38*k).

__Other bases : See note 5 below.__
II) How do I proceed to state these conjectures from the tables?
To illustrate the difficulty of stating these conjectures, I suggest you look at the attached files named "base_x_mat" and "base_x_matcons".

First we had to find the way to display the data in the easiest way possible.

"base_x_mat" shows the occurrences of all the prime numbers appearing in the whole database, specifying in which sequence they appear and at which indexes.

"base_x_matcons" is an extract from "base_x_mat" and shows only those prime numbers that appear at consecutive indexes starting from 1 (or more rarely 0 or 2) for a given sequence.

Attached are the file pairs of base 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17 (the last two files base_15_mat and base_17_mat are missing for reasons of size allowed on the forum).

For example, when we see this extract from the "base_3_matcons" file :

Code:

prime 79 in sequence 3^78 at index 1 2 3
prime 79 in sequence 3^156 at index 1 2 3
prime 79 in sequence 3^234 at index 1 2 3

The conjecture is immediately deduced from this (40).

But it is much more difficult to deduce conjectures (5), (6), (7) and (8) from this excerpt from the "base_2_matcons" file.

Code:

prime 5 in sequence 2^4 at index 1
prime 5 in sequence 2^8 at index 1
prime 5 in sequence 2^12 at index 1
prime 5 in sequence 2^16 at index 1
prime 5 in sequence 2^20 at index 1
prime 5 in sequence 2^24 at index 1
prime 5 in sequence 2^28 at index 1 2
prime 5 in sequence 2^32 at index 1
prime 5 in sequence 2^36 at index 1 2 3 4 5 6
prime 5 in sequence 2^40 at index 1
prime 5 in sequence 2^44 at index 1 2
prime 5 in sequence 2^48 at index 1
prime 5 in sequence 2^52 at index 1
prime 5 in sequence 2^56 at index 1 2 3
prime 5 in sequence 2^60 at index 1
prime 5 in sequence 2^64 at index 1
prime 5 in sequence 2^68 at index 1
prime 5 in sequence 2^72 at index 1 2 3 4 5
prime 5 in sequence 2^76 at index 1 2 3
prime 5 in sequence 2^80 at index 1
prime 5 in sequence 2^84 at index 1 2
prime 5 in sequence 2^88 at index 1 2
prime 5 in sequence 2^92 at index 1 2
prime 5 in sequence 2^96 at index 1
prime 5 in sequence 2^100 at index 1
prime 5 in sequence 2^104 at index 1
prime 5 in sequence 2^108 at index 1 2 3 4 5 6 7 8
prime 5 in sequence 2^112 at index 1 2
prime 5 in sequence 2^116 at index 1 2
prime 5 in sequence 2^120 at index 1
prime 5 in sequence 2^124 at index 1
prime 5 in sequence 2^128 at index 1
prime 5 in sequence 2^132 at index 1 2 3 4 5 6
prime 5 in sequence 2^136 at index 1 2 3
prime 5 in sequence 2^140 at index 1 2 3 4
prime 5 in sequence 2^144 at index 1 2 3 4 5 6 7 8 9 10 11 12 13
prime 5 in sequence 2^148 at index 1 2
prime 5 in sequence 2^152 at index 1 2 3 4 5 6 7 8 9 10
prime 5 in sequence 2^156 at index 1 2 3 4
prime 5 in sequence 2^160 at index 1
prime 5 in sequence 2^164 at index 1 2 3 4 5
prime 5 in sequence 2^168 at index 1 2 3 4 5 6
prime 5 in sequence 2^172 at index 1 2 3 4 5
prime 5 in sequence 2^176 at index 1 2 3 4 5 6 7
prime 5 in sequence 2^180 at index 1 2 3 4 5
prime 5 in sequence 2^184 at index 1 2 3
prime 5 in sequence 2^188 at index 1 2
prime 5 in sequence 2^192 at index 1
prime 5 in sequence 2^196 at index 1 2
prime 5 in sequence 2^200 at index 1
prime 5 in sequence 2^204 at index 1 2 3 4
prime 5 in sequence 2^208 at index 1
prime 5 in sequence 2^212 at index 1 2 3 4
prime 5 in sequence 2^216 at index 1 2 3 4 5 6
prime 5 in sequence 2^220 at index 1 2
prime 5 in sequence 2^224 at index 1 2 3 4 5
prime 5 in sequence 2^228 at index 1 2 3 4 5
prime 5 in sequence 2^232 at index 1 2
prime 5 in sequence 2^236 at index 1 2 3
prime 5 in sequence 2^240 at index 1
prime 5 in sequence 2^244 at index 1 2 3 4 5
prime 5 in sequence 2^248 at index 1
prime 5 in sequence 2^252 at index 1 2 3 4 5
prime 5 in sequence 2^256 at index 1
prime 5 in sequence 2^260 at index 1
prime 5 in sequence 2^264 at index 1 2 3 4
prime 5 in sequence 2^268 at index 1 2
prime 5 in sequence 2^272 at index 1 2 3 4 5 6
prime 5 in sequence 2^276 at index 1 2 3
prime 5 in sequence 2^280 at index 1 2
prime 5 in sequence 2^284 at index 1 2 3
prime 5 in sequence 2^288 at index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
prime 5 in sequence 2^292 at index 1 2
prime 5 in sequence 2^296 at index 1 2 3 4 5
prime 5 in sequence 2^300 at index 1
prime 5 in sequence 2^304 at index 1 2 3 4 5
prime 5 in sequence 2^308 at index 1 2 3 4 5 6 7
prime 5 in sequence 2^312 at index 1 2 3 4 5
prime 5 in sequence 2^316 at index 1 2
prime 5 in sequence 2^320 at index 1
prime 5 in sequence 2^324 at index 1 2 3 4
prime 5 in sequence 2^328 at index 1 2 3 4
prime 5 in sequence 2^332 at index 1 2 3
prime 5 in sequence 2^336 at index 1 2 3 4 5 6 7
prime 5 in sequence 2^340 at index 1
prime 5 in sequence 2^344 at index 1 2
prime 5 in sequence 2^348 at index 1 2 3 4 5 6
prime 5 in sequence 2^352 at index 1 2 3 4 5
prime 5 in sequence 2^356 at index 1 2
prime 5 in sequence 2^360 at index 1 2 3 4 5 6 7 8 9 10
prime 5 in sequence 2^364 at index 1 2
prime 5 in sequence 2^368 at index 1 2 3 4 5 6 7 8 9
prime 5 in sequence 2^372 at index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
prime 5 in sequence 2^376 at index 1 2 3 4 5 6 7 8 9 10 11 12 13 14
prime 5 in sequence 2^380 at index 1 2 3 4
prime 5 in sequence 2^384 at index 1 2 3 4
prime 5 in sequence 2^388 at index 1 2 3 4 5 6
prime 5 in sequence 2^392 at index 1 2 3
prime 5 in sequence 2^396 at index 1 2 3 4 5 6 7 8 9
prime 5 in sequence 2^400 at index 1
prime 5 in sequence 2^404 at index 1 2 3 4 5 6 7 8
prime 5 in sequence 2^408 at index 1 2
prime 5 in sequence 2^412 at index 1 2 3 4 5 6 7 8 9 10
prime 5 in sequence 2^416 at index 1 2
prime 5 in sequence 2^420 at index 1 2 3
prime 5 in sequence 2^424 at index 1 2 3
prime 5 in sequence 2^428 at index 1 2 3
prime 5 in sequence 2^432 at index 1 2 3 4 5 6 7 8
prime 5 in sequence 2^436 at index 1 2
prime 5 in sequence 2^440 at index 1 2 3 4 5 6 7
prime 5 in sequence 2^444 at index 1 2 3 4 5 6 7
prime 5 in sequence 2^448 at index 1 2 3 4 5
prime 5 in sequence 2^452 at index 1 2
prime 5 in sequence 2^456 at index 1 2 3
prime 5 in sequence 2^460 at index 1 2
prime 5 in sequence 2^464 at index 1 2 3
prime 5 in sequence 2^468 at index 1 2 3 4 5 6 7 8 9
prime 5 in sequence 2^472 at index 1 2 3
prime 5 in sequence 2^476 at index 1 2 3
prime 5 in sequence 2^480 at index 1
prime 5 in sequence 2^484 at index 1 2
prime 5 in sequence 2^488 at index 1 2
prime 5 in sequence 2^492 at index 1 2 3 4 5 6 7 8
prime 5 in sequence 2^496 at index 1 2 3 4 5
prime 5 in sequence 2^500 at index 1
prime 5 in sequence 2^504 at index 1 2 3 4 5
prime 5 in sequence 2^508 at index 1 2 3
prime 5 in sequence 2^512 at index 1
prime 5 in sequence 2^516 at index 1 2 3
prime 5 in sequence 2^520 at index 1
prime 5 in sequence 2^524 at index 1 2 3 4
prime 5 in sequence 2^528 at index 1 2 3 4 5 6 7 8
prime 5 in sequence 2^532 at index 1 2 3 4 5
prime 5 in sequence 2^536 at index 1 2 3 4
prime 5 in sequence 2^540 at index 1 2 3 4 5 6 7

I failed in my attempt to write an algorithm that can automatically find guesses.

I spot them by looking at the files, but after a few hours, my eyes start to sting.

Hence question 1 below.

III) Remarks and questions
__Remark 1:__
My long term goal would have been to find a relationship between the occurrences of prime numbers for the bases p and q (prime p and q) with the occurrences of prime numbers in the compound base p*q.

For example: is there a relationship between the occurrences of prime numbers for bases 3 and 5 and between the occurrences of prime numbers for the base 3*5=15?

__Remark 2:__
All this work is very complicated. It is difficult to judge which conjectures are interesting.

The interest in the end, would be to find occurrences for large prime numbers, to accelerate much the factorizations of the terms of the sequences.

__Remark 3:__
Unfortunately, I did not succeed in exploiting the lines of the tables whose indexes are not consecutive for a prime number in a given sequence.

Is there anything to notice with these lines?

__Remark 4:__
I think Ed's statements (post #364 and post #379) are true. But I don't know how to prove them.

__Remark 5:__
I've only presented in this post the conjectures up to base 17.

Writing conjectures is very constraining and takes a lot of time.

**But if someone has interest in the other bases, just ask me.**

I can join in another post all the tables and continue to write the conjectures properly.
__Remark 6:__

I certainly forgot some conjectures.

I may not have seen the most important ones.

A general conjecture may be obvious!

The whole mode can try to find conjectures from the tables!

Now it's up to you to try your luck...
__Remark 7:__
The work is far from over.

I have other ideas to test !

__Question 1:__
How to write a program that automates conjectures from tables (see example above with the prime number 5 in base 2 and the file "base_2_matcons") ?

__Question 2 :__
How to prove this type of result:

If n=3^(6+12*k), then s(n) is divisible by 2^2*7 ?

This may not be very difficult for a professional mathematician, but I don't know how to do it !

__Question 3 :__
How to prove (or invalidate) otherwise than by calculation, this kind of result (conjectures often noted **): If n=3^(78*k), then s(n), s(s(n)) and s(s(s(n))) are all three divisible by 79? This kind of proof must be much, much more difficult than the one we are talking about in question 2!