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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