Curious property of Mersenne numbers.

[arithmeticae]

At the web page:

http://mipagina.cantv.net/arithmetic/mersenne.htm

you will find a curious property on Mersenne numbers.
Any references on previous works or papers specifically on such particular property will be deeply appreciated.

Many thanks,

Ing. Domingo Gomez Morin
Caracas
Venezuela

[arithmeticae]

Any comments on this matter will be of some help, even to say that this is a naive stuff worth of no comments, otherwise: comments, references, articles, papers, anything...


Respectfully. Many thanks

http://mipagina.cantv.net/arithmetic/mersenne.htm


Domingo Gomez Morin

[wblipp]

Domingo,

These are fun observations, but they are curios that do not lead to any generalizations.

1. The second formula is wrong. The denominator should be 10²ⁱ, not 10⁶ⁱ.

2. Factoring out the constant term in the sum leaves geometric series with ratios (2/10⁶) and (2/10²). Using the standard formulas for the sum of geometric series, you can show these series sum to
1/(31*127²) and 1/(7²). Getting only Mersenne numbers in these factorizations is why the observations work.

3. To get a generalization, you need (10^k-2)/2 to factor into Mersenne numbers in general. They don't.

William

[arithmeticae]

Originally Posted by wblipp

Domingo,

These are fun observations, but they are curios that do not lead to any generalizations.

1. The second formula is wrong. The denominator should be 10²ⁱ, not 10⁶ⁱ.

2. Factoring out the constant term in the sum leaves geometric series with ratios (2/10⁶) and (2/10²). Using the standard formulas for the sum of geometric series, you can show these series sum to
1/(31*127²) and 1/(7²). Getting only Mersenne numbers in these factorizations is why the observations work.

3. To get a generalization, you need (10^k-2)/2 to factor into Mersenne numbers in general. They don't.

William

Thanks William, yes you are write, the exponent in the second expression should be 2i, it was just a typo from mine, my apologies.
I have updated the webpage:
http://mipagina.cantv.net/arithmetic/mersenne.htm

Please take a look there again for some additional comments from mine, I hope they can be of some interest to you.

On the other hand, yes, to think about generalizing such particular expressions might not be the right way of thinking, but from my point of view they seems to suggest the existence of other digits-distribution functions for other rational expressions of Mersenne numbers.

Yes, In the case of (1/127^2), the summatory of this geometric series gives as the final result: M5 (M7)^2=499999 showing that my expression was right, but it tells nothing about the amazing digits-distribution in this particular connection between M7 and M5.

I don't know if this is just a fault of mine, but more than a curiousity, i find truly amazing this strange decimal-connection between M7 and M5.

At the webpage please notice what happens at the end of the period of the decimal fraction (length period: 5335, not checked).
Isn't amazing?

One can find a similar expression for generating backwards all the digits.

I think another important point of my message was about any precedents on the study of such digits-distribution for decimal fractions of Mersenne numbers.
¿Is there someone who know about any previous works on such digit-distribution functions?


Domingo Gomez Morin

[arithmeticae]

sorry, another typo: "yes you are write"

it should say: "yes you are right"

sorry

[wblipp]

While there is a generalization, it has nothing to do with Mersenne numbers. Here are examples of the generalization:

Look at the decimal expansion of (1/1724137931). You will find blocks of 11 digits that have the values 2*29, 4*29, 8*29, 16*29 etc. The same doubling phenomenon you observed, but no Mersenne numbers in evidence.

How did I find this? I experimented with (10ⁿ-2)/2 until I found an n with a small factor. For n=11 this is 29 * 1724137931

For another example, look at the decimal expansion of 1/7142857142857. This one will be blocks of 14 digits with value of 2*7, 4*7, 8*7, 16*7, etc. This is because (10¹⁴-2)/2 = 7 * 7142857142857

Or 1/499999999999999, which is blocks of 15 with values 2, 4, 8, 16, etc. because (10¹⁵-2)/2 = 1 * 499999999999999

Or 1/2631578947368421 with is blocks of 17 with values 2*19, 4*19, 8*19, etc because (10¹⁷-2)/2 = 19 * 2631578947368421