the sum of two arbitrary different numbers is always coprime with the sum of three arbitrary different…

Does there exist a sequence consists of infinite positive integers $$a_1,a_2,…$$ such that the sum of two arbitrary different numbers in the sequence is always coprime with the sum of three arbitrary different numbers in the sequence?

Assume that there exist such sequence.

For every prime $$p$$, if there exist three numbers from the sequence $$a_i,a_j,a_k$$ that are divisible by $$p$$, then $$a_i+a_j$$ is not coprime with $$a_i+a_j+a_k$$, contradiction. Thus $$p$$ can divide at most two numbers from the sequence. Therefore there are at most two even numbers from the sequence and there are infinite odd numbers from that sequence.

However, if there is at least one even numbers, then the sum of two odd numbers is not coprime with the sum of two odd numbers and an even one, since both of the sum are greater than $$2$$ and are both even. Hence there are no even numbers in the sequence.

Here I am stuck. How can I progress ? Is there a better way to solve the problem ?

(My other approach is to consider numbers of the form $$m^n+1$$, but it seems hopeless.)

Determine Threshold for different detection test

I am looking at a couple of question based on the image below:

The question then requires I select all thresholds (vertical lines) which could be used for Maximum Likelihood Testing, Maximum A Posteriori Testing (MAP), Bayes Test and the Neyman Pearson test. I also need to select all thresholds which could be used for always deciding H_1 and H_0. Not quite sure of how to approach this question, but the following explains my approach so far.

Now for Maximum Likelihood Testing, the prior probabilities are assumed to be equal and since the question states that one is greater than the other, I have opted to select that none of the vertical can be used as the decision threshold for the Maximum Likelihood Testing. I am not sure about this.

But what approach should I take for the other test? I have thought about finding the likelihood ratio for each vertical line but after that I am unsure of what my next step would be. I would appreciate any advice.

different homepages and one main homepage

I want to create on one website different divisions each with its ows menu and homepage.

in other words”
I you enter the website that will be the Main Homepage.

when click on each of the pictures that have to link to a complete new (own) menu , something like: Home – Milonga – Workshops – Main homepage

how can I fix this?
is it a special plug in, or easy to create without?

How hard would it be for humans to colonize an exoplanet with a size significantly different than that of…

If you look at this list of potentially habitable exoplanets, very few of them actually come close to having the same size as Earth, this means most of them would have surfaces gravities significantly different from those on Earth, and mostly likely significantly different atmospheric pressures as well.

For simplicity’s sake, let’s set aside the issue of whether they need to be terraformed or not, and whether they have native lifeforms.

I’m curious about the problems caused by different gravities:

1. Are humans resilient enough that they can adapt to those different gravities and gravity won’t be an issue?
2. Will it be necessary to genetically modify humans to be able to survive in those different gravities?
3. Will we have to create artificial environments to adjust the gravity and the atmosphere for humans and if an exoplanet has a size significantly different than that of earth, then is it inevitable that it will need artificial environments if people are ever going to colonize it?

Which of (1), (2) or (3) is more likely, given current scientific knowledge?

Did the word “than” mean something different in The Wealth of Nations? [on hold]

What does the last phrase after the last comma in this sentence mean?

And, in the end, the civil magistrate will find that he has dearly
paid for his intended frugality, in saving a fixed establishment for
the priests; and that, in reality, the most decent and advantageous
composition, which he can make with the spiritual guides, is to bribe
their indolence, by assigning stated salaries to their profession, and
rendering it superfluous for them to be farther active, than merely to
prevent their flock from straying in quest of new pastors.

Could the word “than” be replaced with the word “and” to mean what Smith intended?

It seems that the rendering of something superfluous would be along the lines of merely preventing something. But I think of the word “than” to refer to contrasting two elements. Was the word “than” a typo? If not, what did Adam Smith mean? Or did the word “than” mean something different back in Adam Smith’s day?

Why does the text of Shino-sensei’s math problems look different from normal Japanese?

In episode 5 of Boruto, we see Shino-sensei testing Boruto on “Ninja Academy Math” or something along those lines, but when I look closely at the texts, they don’t seem like the usual Japanese characters to me.

I admit that my Japanese is not good but from the numerous anime that I have watched, I feel like I can differentiate the more commonly used kanji when I see them. But then I noticed that none of the ones there are distinguishable to me. Does this have to do with Japanese math and numbers?

DES – encrypt 3 times – DES(DES(DES, k), k’), k) – stronger than 2DES with 2 different keys?

What if were to use DES in the following way:

For a given plaintext `x` and key `k` we will perform: `DES(DES(DES, k), k'), k)`, where `k'` is the complementary of `k`.

If an adversary has a small amount of text and their ciphertexts, would he still have to run 3 * 2^56 encryptions: for each key perform `DES(DES(DES, k), k'), k)` and therefore this encryption is stronger than 2DES with 2 keys (meet in the middle requires 2^57 runs) but less than real 3DES which is much more than 2^57 ?

Or is there a smarter way ?

Question: If you lost medicaid from one state, is it hard to get a different state’s medicaid?

I don’t want to sound like I’m complaining about getting myself health benefits, but I’m diagnosed with epilepsy and I’m trying to move to a different state for a job offering, but my medical expenses are just far above my pay stub, and I’m not sure if it’s the right decision to move. career vs health,

Thank you, guys
Good day.

Why don’t we play different openings in chess?

Chess comes in many forms, both with respect to the clock and the board format.

But why don’t we have chess tournaments where we play different openings? That is, a certain list of openings is picked (it could be publically announced prior to the tournament, or perhaps a surprise reveal), and then both players play games with those openings (each opening is played twice, once as black, once as white).

This seems like quite a fun thing to do and an obvious extension to the many forms of chess we already have?

What’s the drawback?