Mersenne exponents are connected through a simple binary operation tree

[Martin Doina]

Every Mersenne exponent (prime `p` where `2^p - 1` is prime) can be generated by starting from a seed (2 for Family A, 3 for Family B) and repeatedly applying two operations based on a binary pattern:

| Operation | Meaning |

|--------------|-------------|

| 0 | Multiply by 2: `n → n × 2` |

| 1 | Multiply by 2 and add 1: `n → n × 2 + 1` |

Example: How M10 (89) is Generated

```

Seed: 2

Pattern: 1 1 0 0 1

M10: 89 (pattern: 11001)

Sequence: 2 → 5 → 11 → 22 → 44 → 89

Step 1: Start at 2

Apply '1': 2 × 2 + 1 = 5

Apply '1': 5 × 2 + 1 = 11

Apply '0': 11 × 2 = 22

Apply '0': 22 × 2 = 44

Apply '1': 44 × 2 + 1 = 89

Small Mersenne exponents act as way points - numbers like 2, 3, 5, 7, 13, 17, 19, 31 appear as intermediate steps in many sequences

Introduction:

The Classification System

I've organized this into:

- Family A: Starts with seed 2

- Family B: Starts with seed 3

- Groups: Based on the first two binary digits (00, 01, 10, 11)

- Subgroups: Based on the first three binary digits (000, 001, 010, 011, 100, 101, 110, 111)

Why Is This Important?

This is a generative classification - instead of just listing Mersenne exponents, I've found a structural relationship between them.

Your opinion on this?

Extended in my orcid https://orcid.org/0009-0002-3855-2268

[Allan Wechsler]

Are you, then, in a position to predict the next Mersenne exponent, or to give advice on classes of candidate exponents to give special attention to?

-- Allan

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[Martin Doina]

Hi Alan, I am still researching but I have candidates; 

118692143: "1000100110001100100101111",

129077551: 1101100011001000100101111

178664021: "10101001100011001001010101",

6671927551: "0001101101011011000100011111111"

Pattern: 1000100110001100100101111

25 Length 12 Legs (1's) 13 Shells (0's) 8 Digital Root
118692143
3 → 7 (×2+1) → 14 (×2) → 28 (×2) → 56 (×2) → 113 (×2+1) → 226 (×2) → 452 (×2) → 905 (×2+1) → 1811 (×2+1) → 3622 (×2) → 7244 (×2) → 14488 (×2) → 28977 (×2+1) → 57955 (×2+1) → 115910 (×2) → 231820 (×2) → 463641 (×2+1) → 927282 (×2) → 1854564 (×2) → 3709129 (×2+1) → 7418258 (×2) → 14836517 (×2+1) → 29673035 (×2+1) → 59346071 (×2+1) → 118692143 (×2+1)

Pattern: 1101100011001000100101111
25 Length 13 Legs (1's) 12 Shells (0's) 1 Digital Root
129077551
3 → 7 (×2+1) → 15 (×2+1) → 30 (×2) → 61 (×2+1) → 123 (×2+1) → 246 (×2) → 492 (×2) → 984 (×2) → 1969 (×2+1) → 3939 (×2+1) → 7878 (×2) → 15756 (×2) → 31513 (×2+1) → 63026 (×2) → 126052 (×2) → 252104 (×2) → 504209 (×2+1) → 1008418 (×2) → 2016836 (×2) → 4033673 (×2+1) → 8067346 (×2) → 16134693 (×2+1) → 32269387 (×2+1) → 64538775 (×2+1) → 129077551 (×2+1)

---

Pattern: 10101001100011001001010101
26 Length 12 Legs (1's) 14 Shells (0's) 8 Digital Root
178664021
2 → 5 (×2+1) → 10 (×2) → 21 (×2+1) → 42 (×2) → 85 (×2+1) → 170 (×2) → 340 (×2) → 681 (×2+1) → 1363 (×2+1) → 2726 (×2) → 5452 (×2) → 10904 (×2) → 21809 (×2+1) → 43619 (×2+1) → 87238 (×2) → 174476 (×2) → 348953 (×2+1) → 697906 (×2) → 1395812 (×2) → 2791625 (×2+1) → 5583250 (×2) → 11166501 (×2+1) → 22333002 (×2) → 44666005 (×2+1) → 89332010 (×2) → 178664021 (×2+1)

-----------

6671927551

Lattice 1 pattern: 0001101101011011000100011111111

3 → 6×2 → 12×2 → 24×2 → 49×2+1 → 99×2+1 → 198×2 → 397×2+1 → 795×2+1 → 1590×2 → 3181×2+1 → 6362×2 → 12725×2+1 → 25451×2+1 → 50902×2 → 101805×2+1 → 203611×2+1 → 407222×2 → 814444×2 → 1628888×2 → 3257777×2+1 → 6515554×2 → 13031108×2 → 26062216×2 → 52124433×2+1 → 104248867×2+1 → 208497735×2+1 → 416995471×2+1 → 833990943×2+1 → 1667981887×2+1 → 3335963775×2+1 → 6671927551×2+1

In my github there are some apps that helps:

https://github.com/gatanegro/MERSENNE-COLLATZ/tree/main/MERSENNES%20EXPONENTS%20CONSTRUCTION%20FROM%20BINARY%20PATTTERN

I am into find how to build the pattern, I find some rules but I have to test more ...

Thanks for email

Doina

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[Allan Wechsler]

The Great Internet Mersenne Prime Search (GIMPS) project, at mersenne.org, reports on their front page that all exponents below 139,760,749 "have been tested at least once".

An explicit factor of 2^129,077,551 - 1, 5387100730550510599, was found in early 2009.

Furthermore, 2^118,692,143 - 1 was proven composite in late 2023, though nobody has reported an explicit factor.

195498102386663 is an explicit factor of 2^178664021 - 1, found in late 2007.

But your last candidate exponent, 6671927551, has not been tested yet. If you download the GIMPS software, you can use it to test specific exponents; there is also an interface for suggesting exponents of potential interest. I encourage you to do this.

Do not be discouraged. If you have a candidate generator that succeeds even once in a thousand guesses, you are doing better than the existing state of the art. 

-- Allan

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[David desJardins]

On Fri, Mar 27, 2026 at 5:55 PM Martin Doina <dhela...@gmail.com> wrote:

Every Mersenne exponent (prime `p` where `2^p - 1` is prime) can be generated by starting from a seed (2 for Family A, 3 for Family B) and repeatedly applying two operations based on a binary pattern:

| Operation | Meaning |

|--------------|-------------|

| 0 | Multiply by 2: `n → n × 2` |

| 1 | Multiply by 2 and add 1: `n → n × 2 + 1` |

Every integer larger than 1 can be generated by starting with 2 or 3 and then repeatedly applying these two operations! 

[Martin Doina]

True, every integer larger than 1 can be generated by starting with 2 or 3 and then repeatedly applying these two operations, but we observe that smallest Mersennes exponents are nested from start into largest ones (fractal  -shared branch) and the terms length have pattern too, also after  duplicate some small numbers  follow than a unique path.Each sequence contain at least one Mersennes exponent small.

[Martin Doina]

Allan , I am not interested in finding new Mersennes but to find the rules through patterns. The last one I get is 

422103702739

37 Depth, 19 legs, 18 shells

pattern: 0001001000111010101110101110011010011

3 → 6×2 → 12×2 → 24×2 → 49×2+1 → 98×2 → 196×2 → 393×2+1 → 786×2 → 1572×2 → 3144×2 → 6289×2+1 → 12579×2+1 → 25159×2+1 → 50318×2 → 100637×2+1 → 201274×2 → 402549×2+1 → 805098×2 → 1610197×2+1 → 3220395×2+1 → 6440791×2+1 → 12881582×2 → 25763165×2+1 → 51526330×2 → 103052661×2+1 → 206105323×2+1 → 412210647×2+1 → 824421294×2 → 1648842588×2 → 3297685177×2+1 → 6595370355×2+1 → 13190740710×2 → 26381481421×2+1 → 52762962842×2 → 105525925684×2 → 211051851369×2+1 → 422103702739×2+1

In a few seconds, my software avoids duplicates and after many composite we hit a possible prime. Gimps framework can be good till a certain size number , we need to find a new approach to filter at least before testing everything. And Gimps test their programmed numbers not  what one proposes. WE see that the greatest M contain the smallest ones but we can not see the path length from which family start.

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[Daniel Mondot]

All your process is doing is taking the mersenne exponent (here 89), converting it into binary (1011001) and removing the leading 2 digits which can only be 10 for your family A or 11 for your family B.

You don't need to explicitly show this elaborate series of (x2) and (x2+1) operations, it's just a binary decomposition.

And I don't see how you can predict anything from this process.

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[Gareth McCaughan]

On 28/03/2026 11:00, Martin Doina wrote:
> Allan , I am not interested in finding new Mersennes but to find the
> rules through patterns.

What Allan has been too polite to say is: Your alleged patterns appear
to contain no actual new information beyond just saying "we can write
the exponents as binary numbers", but if they _do_ have new information
in then it should be possible to verify this by using them to find new
Mersenne primes more efficiently than just by picking large primes for
their exponents.

(Also: your comments about these suggest that you are making substantial
use of LLMs for this. Some day the AIs will be better at mathematics
than the best humans, no doubt, but that day is not yet here, and
unfortunately one thing LLMs are very good at is persuading humans that
they-plus-the-LLMs have come up with something new and interesting when
actually they have not. It's not _impossible_ that you have found
something important by working together with an LLM, but empirically
almost all people who think they have done are mistaken.)

--
g

[Martin Doina]

What I am doing is to find the link between patterns :

attice 6 (Pattern: 111)

 Contains duplicates
3 → 7  → 15  → 31 
         |pattern: 1 (15 → 31)
         |pattern: 01 (15 → 30 → 61) We get next M with previous pattern. 
         |pattern: 111 (15 → 31 → 63 → 127) We get next M with previous pattern. 

I use my apps not the AI . We can get the largest M using the patterns of the smallest. Th Mersennes exponents are not random , must be a rule ... 

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