nt.number theory – Question about odd positive integers and Fermat factors

Sequence $ 17, 257, 641, 65537, … $ It consists of odd positive integers that match these three conditions:

  1. The period duration of the decimal expansion of $ 1 / n = 2 ^ x $ and divide $ n-1 $.
  2. The sum of $ n = 2 ^ x $.
  3. The cycle duration of $ n = 2 ^ x $.

($ x $ it's a positive integer)

(using $ n = $ 23 as an example to define the sum of $ n $ and the cycle duration of $ n $):

Step 1: Get the strange part of $ 23 + ~~ $ 1, which $ ~~ $ 3,$ ~~ 3 times2 ^ 3 = 23 + ~~ 1 $,obtain $ s_1 = $ 3
Step 2: Get the strange part of $ 23 + ~~ $ 3, which $ 13 $,$ 13 times2 ^ 1 = 23 + ~~ 3 $,obtain $ s_2 = 1 $
Step 3: Get the strange part of $ 23 + $ 13, which $ ~~ $ 9,$ ~~ 9 times2 ^ 2 = 23 + 13 $,obtain $ s_3 = 2 $
Step 4: Get the strange part of $ 23 + ~~ $ 9, which $ ~~ $ 1,$ ~~ 1 times2 ^ 5 = 23 + ~~ 9 $,obtain $ s_4 = 5 $

Continuing this operation (with $ 23 + $ 1) Repeat the same steps as above.
exist $ 4 $ steps in the cycle, so the cycle duration of $ 23 $ is $ 4 $, And the sum of $ 23 $ is $ s_1 + s_2 + s_3 + s_4 = $ 11.

It seems that all known elements of the sequence are Fermat factors, how is that?

Group dynamics: practical use of the O card, or how to measure positive consent on the fly

I'm getting ready to run the game of Bluebeard's Bride with a couple of players that I don't know very well. This game can be quite heavy in disturbing content, so I certainly plan to have an equivalent X card in the game. At the same time, I have given a clear indication of the subject and some of the possible triggers in the pre-game propaganda (it will be executed in a small convention dedicated exclusively to role-playing games), so it is assumed that the players will be willing to Experiment at least a little and push your limits.

As it is a single session game, in a predefined time frame (5-6 hours in total), there is a limit on the amount of research / questionnaires & # 39; session 0 & # 39; What can I do before the game? I also don't expect to have any contact with the players before the game itself.

I am seriously considering having the equivalent of the O card in addition to the X card. For people who are not familiar with the term, here is a definition of TTRPG Safety Toolkit

The O card can be used at any time if a
The participant wants to continue with the content.
When using the card OR by touching the card
or by typing an "O" in the chat, the group is fine for
Continue with the content. They can also regularly
be driven by an "O," he asked aloud or in the
Chat to register if everyone is still well.

Let's ignore the game online.

How does it work in practice with multiple players? The X card is simple: a player is rescued, the scene stops. But with the O-card, is it enough that the player directly involved touches a card to increase / follow the narration and rest? Can you X-card if you disagree? Can other players use O-card, even if they are only listening to atm? Or do we make a quick vote, which can be quite uncomfortable with 5 players and exert a kind of group pressure on the last one that does not join, which are those techniques intended to avoid?

With LARPs it is a little easier with red / yellow / green safety words, because

  • you often interact with only one person who may be affected by your actions
  • you often ask about physical interaction but you use verbal confirmation, which gets less intrusive in the flow

In the TTRPG, the physical gesture on the X card provides the same distinction between the action (which is verbal) and the security mechanism (tap in this case): verbal consent techniques would be more invasive.

Do you have any other technique for players to indicate their consent to move to a higher march & # 39; On the fly, which work with 5 players?

linear algebra: semi-defined positive symmetric matrix $ B $, shows that there is a $ A $ s.t. $ B = AA ^ T $

If there is a positive symmetric / semi-defined matrix $ B in R ^ {n times n} $ that has rank $ r $.

How do I prove that there is a $ A en R ^ {n times r} $ such that $ B = AA T

To be positive semi-defined, I know that the following inequality has to be maintained:

$$ x ^ TBx geq 0 $$

elementary number theory: assuming that for each integer n> 1 there is a cousin between n and 2n, show that each positive integer can be written as the sum of different prime numbers

Note that in this problem 1 is treated as a cousin

My test is like that

Let's say we want the number $ q $ be written as sum of cousins.$ q $ It can be odd or even. First consider the case when q is even.

Assuming there is an excellent word $ p_1 $ such that

$ frac {q} {2} <p_1 <q $ now we should add $ q-p_1 $ to $ p_1 $
Arrive $ q $ Realise $ q-p_1 $ it's odd

Then there is a prime number say $ p_2 $ such that

$ frac {q-p_1-1} {2} <p_2 <q-p_1 $

Yes now$ p_1 + p_2 = q $ We stop here otherwise we continue this process. One more step if $ p_1 + p_2 $ not equal to $ q $.

Then there is a first say p_3 such that

$ frac {q-p_1-p_2} {2} <p_3 <q-p_1-p_2 $

We continue this process until we reach $ q $Notice that at each step the value is decreasing $ p_1> p_2> p_3 $ Then this process ends at some point. We can do the same when it is odd. This completes the test.

Is this proof legitimate or is there a hole in it?

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additive combinatorics: is any set of positive densities of positive integers almost rational?

(All sets in this discussion are sets of positive integers.) I say a set $ V $ It is "rational" if it meets one of the following conditions:

I. $ V $ it is finite (or empty);

II $ V $ it can be written as the union of a finite set (possibly empty) and finitely many arithmetic progressions of infinite length.

Leaving $ Delta $ denotes the operation of symmetric difference, I say a set $ W $ it is "almost rational" if there is a rational set $ V $ such that $ V Delta W $ It has zero natural density. Clearly every $ W $ of density $ 0 $ or $ 1 $ it's almost rational (with $ V $ any finite set being in the first case, and $ V $ being any set whose complement is finite, in the latter case). My question is: is there a set of positive natural density that is no almost rational?

Google warning: the value in the "duration" property must be positive

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What are the positive effects of social networks?

Hello friends,

What are the positive effects of social networks?

Best practice to affirm a positive IF condition

What's better: leave the IF block blank to do nothing, or to add a statement that basically does nothing (each link already has a href attribute)?

export function fixRelativeLinksOutsideOfEpub(dom): void {
    const content = dom.window.document.querySelectorAll('a');
    content.forEach((element: HTMLElement) => {
        const href = element.getAttribute('href');
        if (href.indexOf('#') === 0) {
            // do nothing
        } else {
            element.setAttribute('target', '_blank');
export function fixRelativeLinksOutsideOfEpub(dom): void {
    const content = dom.window.document.querySelectorAll('a');
    content.forEach((element: HTMLElement) => {
        const href = element.getAttribute('href');
        if (href.indexOf('#') === 0) {
            element.setAttribute('href', `${href}`);
        } else {
            element.setAttribute('target', '_blank');

Mathematics education: is -1 ^ 2 positive or negative?

This may be a fairly basic question, but I really can't find an answer on this topic. The calculator says one thing, but I see that the teachers say the other: Is -1 ^ 2 positive or negative?

According to the calculator:

-1 ^ 2 is negative since the exponents go before the negative signs; – (eleven).

(-1) ^ 2 is positive since parentheses come before exponents; (-eleven).

Now that we have that part out of the way …

When someone says -1 ^ 2, should they perceive it as 1 negative or positive?

I was watching a video that updated my algeba when I noticed that it was perceived as (-1) ^ 2, positive (otherwise, the answer would be incorrect) when my initial reaction would be -1 ^ 2 and it would be negative depending on the way it was written.

https://www.youtube.com/watch?v=P8W2M0jq2Qs&list=PLybg94GvOJ9FoUSDODs14ck3OzSje4WZb&index=14 (at the 1:00 minute mark)