# Tagged Questions

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### Resources about infinite primes of form $n^2 + 1$

Where can one find existing work on the following problem? Prove there are infinitely many primes of the form $n^2 + 1$. Resources about related work would also be appreciated.
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### References for Legendre's prime-counting function

This question is about Legendre's prime-counting function, the one that can be used to calculate the exact amount of prime numbers that are less than or equal to a given number (as long as the number ...
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### Is a Lucas Number with either a power of 2 or a prime index always coprime with all previous Lucas Numbers?

I was looking at this webpage which lists the first 200 Lucas Numbers color-coded with their prime factors and I noticed that all the Lucas numbers with power of two or prime indexes were relatively ...
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### What's the Shannon entropy of the prime numbers?

Here's a note that calculates it as 1. Do you know of any other calculations? http://www.math-math.com/2014/05/shannon-entropy-shannon-entropy-of.html
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### Is there any prize for proving conjecture on Fermat's prime ?-+

I know this site is for mathematical questions and answer places, but I need a little help from you in some other aspect. I have searched in google but didn't get any satisfactory answer for it. This ...
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### A question about an asymptotic formula

I've been told that the asymptotic formula $\pi(x+y)-\pi(x)\sim y/\ln x$ holds for $y\ge x^{1/2+\varepsilon}$ if Riemann's hypothesis is true, but I was unable to find a journal reference for this. ...
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### “Dirichlet's theorem” on pairs of consecutive primes

The number of primes in each of the $\phi(n)$ residue classes relatively prime to $n$ are known to occur with asymptotically equal frequency (following from the proof of the Prime Number Theorem). ...
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### Is there a function that only generates primes?

The title sums it up: does there exist a "nice" injective function $f(n)$ such that $f(n)\in\mathbb P$ for all $n\in\mathbb N$? I'm having difficulty specifying exactly what I want "nice" to mean, ...
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### Questions about assigning a probability to a randomly chosen large integer $n$ being prime

I heard this question a few days ago, so reciting from memory: If I were to randomly choose an arbitrarily large positive integer $n$, could I write a function that determines the likelihood of it ...
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### $\sum_p z^p$ where $p$ is prime

I've started reading Shakarchi's Complex Analysis, and I thought about something interesting. If I haven't mistaken, for any subsequence $A\subset \mathbb{Z}^+$, $\sum_{n\in A} z^n$ has radius of ...
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### Matiyasevich polynomial proof

Can someone provide a proof, or a link to a proof, of why does the Matiyasevich polynomial always generate primes for the nonnegative results? Any help will be appreciated.
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### Longest Odd/Even Sequence in Composite Patterns

NOTE I have completely reworded this because I made a complete hash of it the first time, it got worse as I added to it. I apologize to anyone who might have been confused, and hope that this will be ...
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### Pollard-Strassen Algorithm

I'm aware that the Pollard-Strassen algorithm can be used to find all prime factors of $n$ not exceeding $B$ in $O\big(n^{\epsilon} B^{1/2}\big)$ time. This is really useful because I need to find all ...
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### Parameters giving maximal-length Collatz-like sequence

In a recent question the following recursive sequence was considered: $$a_{n+1} = \cases{\frac{a_n}{2} & a_n is even \\ a_n +d & a_n is odd}, \quad a_1 = d + 1$$ where $d$ is an odd ...
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### infinitely many prime numbers with prescribed digits

My main question is the generalization, though one can answer the first one and it will get accepted. Are there infinitely many primes involving $3,7$ only? Generalization: For what sets of given ...
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### Primes of the form $\lfloor x^k\rfloor$

I'm looking for a result (embarrassingly enough, a somewhat famous result) which shows the infinitude in some sense I don't recall of primes of the form $$\lfloor x^k\rfloor$$ for $k$ fixed and ...
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### A good introduction to Prime Numbers

I'm looking for a good introduction to Primes Numbers, their properties, and some of the better known theorems concerning them. I would prefer references assume knowledge of undergraduate level real ...
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### Discussion on twin prime conjecture

I understand where I am wrong in my previous post. Also, I am very thankful to all members, who answered and showed my errors in post. Now, I would like to know the proof for the following. "The ...
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### A trivial but maybe nonetheless non-trivial method of inferring primality

The topologist J. H. C. Whitehead (not to be confused with his famous uncle) said it is naive to think a theorem is trivial merely because its proof is trivial. Hence I'm wondering if a certain ...
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### Primes of the form $n^2+1$ - hard?
I met a student that is trying to prove for fun that there are infinitely many primes of the form $n^2+1$. I tried to tell him it's a hard problem, but I lack references. Is there a paper/book ...
Has work been done on looking at what happens to the exponents of the prime factorization of a number $n$ as compared to $n+1$? I am looking for published material or otherwise. For example, let ...