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Old 2020-08-04, 08:42   #384
garambois
 
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Oct 2011

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I think I've come up with a new conjecture again.
I don't know if it's already known, please let me know if it is ?


This new conjecture concerns on the other hand only one prime number for base 3, but what is new is that we have the presence of this prime number always in the decomposition of two consecutive terms of the sequence !
Here is the statement of the conjecture :
For any aliquot sequence starting with a number of the form 3^(26*k), k integer, the prime number 398581 always appears in the decomposition of the terms of index 1 and index 2.

This is a small conjecture which concerns only a particular case, but perhaps more general conjectures could be found.

To find this, I proceeded as follows :

1) I found this line in the EdH tables :
Code:
prime 398581 shows up 18 times (26, 52, 78, 104, 130, 156, 182, 208, 234).
18 times and only 9 exponents, this is not "usual" ! 18/9=2.

2) I ran a program that makes the indexes appear and I saw this :
Code:
base 3    prime 398581    exponent 26 at index 1
base 3    prime 398581    exponent 26 at index 2
base 3    prime 398581    exponent 52 at index 1
base 3    prime 398581    exponent 52 at index 2
base 3    prime 398581    exponent 78 at index 1
base 3    prime 398581    exponent 78 at index 2
base 3    prime 398581    exponent 104 at index 1
base 3    prime 398581    exponent 104 at index 2
base 3    prime 398581    exponent 130 at index 1
base 3    prime 398581    exponent 130 at index 2
base 3    prime 398581    exponent 156 at index 1
base 3    prime 398581    exponent 156 at index 2
base 3    prime 398581    exponent 182 at index 1
base 3    prime 398581    exponent 182 at index 2
base 3    prime 398581    exponent 208 at index 1
base 3    prime 398581    exponent 208 at index 2
base 3    prime 398581    exponent 234 at index 1
base 3    prime 398581    exponent 234 at index 2
The conjecture then appeared immediately !


I'd like to try and find some more conjectures like that. We have to look at all the bases. But I'm not sure how to write the programs to spot this kind of case !
Maybe for a base, we have to find cases where the number of occurrences of the prime number is a multiple of the number of exponents for which the prime number appears ? In the example above, we have 18/9=2, so we have two indexes per sequence where the prime number 398581 appears and moreover, these indexes are consecutive !
Are there ratios of 3 (3 consecutive or not consecutive indexes), or more ?
Answer in a few days or weeks... And certainly other unexpected things will appear !
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