Find the first 20 primes found by the classical proof of the infinitude of the set of primes

Begin with P={2}; then form,m, the sum of 1 with the product overall elements of P. Place the smallest prime factor of m into P and repeat.

Suppose p = {p1,p2,…,pr}, then m = 1+ p1p2p3…pr.

Example:

2 is prime and 2+1 = 3 is prime;

2 * 3 +1 = 7 is prime;

2 * 3 * 7 +1 = 43 is prime;

2 * 3 * 7 * 43 +1 = 1807 = 13 * 139, then 13 is the prime;

Thus the first 5 prime number found by the classical proof is {2,3,7,43,13}.

So how to use this proof to find the first 20 prime in Mathematica?
Thank you.

proof writing – New to transitive sets,need help in doing and understanding this stuff

Following problem is from Pinter’s book of set theory

Prove that a is a transitive set iff the
following holds :if B∈CandC∈A,thenB∈A

6.5 Definition A set A is called transitive if, for each x ∈ A, x ⊆ A.

Are B and C transitive sets?

If A$^+$=A$cup${A} could I say B= A$^+$ lexicographically
speaking?

Attempted proof

If A is transitive then x ∈ A, x ⊆ A.
after this I am stuck

metric spaces – Proof that two balls intersect

Prove that the two open balls 𝐵(𝑥,0.9) and 𝐵(𝑦,0.9) which are both contained in 𝐵(0,1) intersect.
I have tried to prove they both contained zero but it doesn’t look like they do and I am getting confused on how to apply the distance function.
We are looking at the metric space (𝑅𝑛,𝑑1,2𝑜𝑟∞)

Does leaving behind a business and child count as proof of intention to return for UK Visa?

I know this has been done many times before but I’m seeking advice on my situation.

I’m a professional British citizen living in Asia. I have been with my Filipino Girlfriend for around a decade. I provide for her financially. She has a small food and beverage business in the Philippines which is in her name. She has all the paperwork to prove this, including business permits etc. Unfortunately, as it’s basically a cash in hand business not much of the takings are deposited in a bank, although she does have an account.

She has a 14 year old Daughter from a previous relationship and we have an 18-month old Son together who has my surname and I am named on the birth certificate.

I would like to take my Girlfriend and Son to the UK for 2 weeks this
year to visit my parents/my sons grandparents who are elderly and
would find a long journey difficult. I have the following questions
surrounding her being able to prove that she intends to return:

  • Would the fact that her Daughter, who lives with her and is reliant
    on her, will be staying behind in the Philippines be enough to show
    she intends to return?
  • Would the fact that she owns her own small business (and has all the
    documentation to prove this) be proof that she intends to return,
    despite not having accounts etc to show it is profitable?

Does the fact that I am my sons Father and I want him to see his elderly grandparents and meet his wider family members count for anything? Obviously an 18-month old baby is too young to spend 10 to 14 days away from his Mother.

For the record I will be sponsoring her trip/paying for flights/paying for private medical insurance for the both etc.

Edited to add from comments: My girlfriends mother will be taking care of the daughter and has provided a letter to state this. I can prove that I support my girlfriend, although I work and live in Singapore.

proof techniques – Prove (p → ¬q) is equivalent to ¬(p ∧ q)

I need to prove the above sequent using natural deduction. I did the first half already i.e. I proved $(prightarrowneg q)rightarrow neg (p wedge q)$, but I’m stuck on where to start for the reverse i.e. proving $neg (p wedge q) rightarrow (prightarrowneg q)$. I figured I would start by assuming $neg (p rightarrow neg q)$ and then working towards a contradiction, but I’m still at a dead end. Can someone point me in the right direction? Thanks.

cryptography – How does Zero Knowledge Proof prevent lying?

I have seen and read several videos and articles that try to explain how Zero Knowledge Proof systems work. In these examples, there is a verifier of information, and a prover of information. The verifier asks the prover to prove that they know some information. Metaphors I have seen usually are:

The prover wants to prove they are not colorblind and gives the verifier two colored items. The verifier is color
blind and switches the items around (or not). The prover can tell if
the items were switched. Repeat this a hundred times successfully and
the prover has proven to the verifier that they really are not colorblind.

I’ve seen variations with Pepsi/Coke taste tests, colored pens or balls, etc. However, this kind of proof requires that the verifier has some sort of information at their disposal already (whether they switched the balls around or not).

I’ve also heard examples such as:

  • A liquor store asks a customer if they are at least of drinking age. The customer can prove they are without having to show their ID.

  • A bank asks a customer to prove they have a minimum balance so that they can get a loan, without having access to the customer’s bank account.

These are examples where the verifier (the liquor store and the bank) doesn’t have the same information at their disposal (such as the example of switching the items around).

If a liquor store asks a customer if they’re of drinking age, what’s preventing the customer from just lying as many times as needed to buy their alcohol?

fa.functional analysis – Trying to recover a proof of the spectral mapping theorem from old thesis/paper with continuous functional calculus

In my research group in functional analysis and operator theory (where we do physics and computer science as well), we saw in an old Russian combination paper/PhD thesis in our library a nice claim about the spectral mapping theorem’s possible proof. Let me attempt to bring the context here. I should mention there are some nice results in this paper that I wanted to use and generalize for my own research, I hope to accurately bring the context below.

They bring up the continuos functional calculus $phi: C(sigma(A)) rightarrow L(H)$ for a bounded, self-adjoint operator on a Hilbert space A. This is an algebraic *-homomorphism from the continuous functions on the spectrum of $A$ to the bounded operators on $H$. The paper’s spectral mapping theorem basically says in this context $$ sigma(phi(f)) =f(sigma(A)) $$ and the paper says something nice about this. It does not actually give a proof but it says there is a nice way to prove it using both inclusions with the inclusion $ f(sigma(A)) subseteq sigma(phi(f)) $ sketched in the following way: the author supposes $ lambda in f(sigma(A)) $ and says “it is very obvious” that there exists a vector $h in H$ with $|h|=1$ such that $|phi(f)-lambda)h|$ is arbitrarily small which shows $lambda in sigma(phi(f))$ which shows the desired inclusion.

The author says that it is “very obvious” to show this but I am a bit stumped. The way I would construct the continuous functional calculus is to start with polynomials and then generalize to $ C(sigma(A)) $ based on the Weierstrass approximation theorem on the real compact set $sigma(A)$ and the BLT theorem. The inclusion $sigma(phi(f)) subseteq f(sigma(A))$ is, I think, quite obvious but the other one in the above context has me stumped. Since I am already working on generalizing some results, I would really love to know how the author proves the inclusion with the method of showing the mentioned vector exists. Maybe use approximation in some way, but even though I suspect it is simple, I still do not see the author’s proposed proof. Can someone here please help me recover it? I thank all interested persons.

proof provenance of funds – I would like to visit the UK in the future. Can someone make sense of my weird bank situation?

My wife and I would like to visit the UK in the future, but I was wondering whether my situation is going to be an issue or not (I would hope not, but you never know).

I am self employed, and I make a nice amount of money working as a contractor. I have two bank accounts: the personal one, and the business one.

The way it works is I can use the funds in my business account same as it were my personal account. There’s no dividend or anything – my government only cares for the taxes I pay as self employed when the time comes. It’s perfectly legal – any money I withdraw doesn’t count as expenses, but anyways.

I usually have a specific day of month (+/- a day) when I transfer a fixed amount of money from the business account to my personal account, which I intend to use as living expenses. Sometimes I have some extra expenses (e.g. unexpected trip to the doctor), which I then again transfer from the business account to the personal one.

I’m wondering if such a setup can make it difficult for me to convince the officer I have enough funds to cover my expenses while in the UK. Can I even use my business account statements as proof of my income?

graphs – Karger algorithm variation to find s-t cut. proof that it is not possible?

Consider executing a variant of the Karger’s algorithm, in
which each time we choose a uniformly random edge e, conditioned on the two (super) nodes which are
endpoints of e do not contain both s,t and we contract e; in other words, we never merge s,t in the course of the algorithm. We repeat this until only two (super) nodes remain where one contains s and
the other contains t. So, we just output the corresponding s-t cut.
In this exercise we want to show this algorithm fails, i.e., the probability of success is so small that
we cannot turn it into a high probability success even by running polynomially many copies of this
algorithm. Construct a graph G with n vertices such that the above algorithm outputs the min s-t cut
with exponentially small probability, namely c−n for some constant c > 0. For convenience, graph G may
also have parallel edges

vectors – Proof of orthogonality using Einstein notation

Prove that $rottextbf{A}$ is a vector orthogonal to $textbf{A}$. First show that $varepsilon_{ijk}k_jk_k=0$.

I have no problem showing the second part but I don’t know how to use it in the proof.
I want to prove the orthogonality with scalar product so I should obtain 0 from the equation. So far I have:
$$nabla timestextbf{A} cdot textbf{A} = hat{e_i}a_i(varepsilon_{jkl}hat{e_j}frac{partial}{partial x_k}a_l)=delta_{ij}varepsilon_{jkl}frac{partial}{partial x_k}a_la_i$$
I don’t know what to do after this so that I use the shown property.