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Prime numbers made of permutations of digits of consecutive positive integers

We knows that no prime of the form $1\_2\_...\_n$ has been found yet... at least for $n$ up to $10^6$ (see https://mathworld.wolfram.com/SmarandachePrime.html). Now, a partial answer to your question ...
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1 vote

Prime numbers made of permutations of digits of consecutive positive integers

My impression is that the prime numbers in $S(3k+1)$ are exactly in the proportion they should be. Note that numbers in $S(3k+1)$ are smaller than $10^{D(3k+1)-1}$ because they have $D(3k+1)$ digits. ...
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2 votes

Priming for the primes

This concerns prime powers and not strictly primes, but the fact that the topological Tverberg conjecture (see e.g. https://arxiv.org/abs/1605.05141) holds for prime power values of “r” and not any ...
2 votes

Priming for the primes

Virtually all important examples of error-correcting codes that have a chance to be optimal and practical in applications utilize prime numbers.
1 vote

Priming for the primes

Prime numbers occur in a major way in the theory of tactical configurations, in particular in finite geometries.
2 votes

Priming for the primes

Feedback with carry shift register sequences (FCSRs) are an arithmetic parallel of the LFSRs in the answer by @WlodAA. They can be represented using $N-$adic numbers and achieve maximal period when $N$...
7 votes

Priming for the primes

Here is a characterization of entropy functions due to Faddeev in 1956 (see pp. 229-231 of Faddeev's paper here if you read Russian or Chapter 1 of A. Feinstein's 1958 book Foundations of information ...
4 votes

Priming for the primes

For me, the most spectacular fact that uses primes in an essential way and which no prime number theorist would be interested in is the fact that Tarski monsters exist for all sufficiently large ...
2 votes

Priming for the primes

The prime numbers come in handy when studying countability. For example, one can prove that the set of finite subsets of $\mathbb{N}=\{1, 2, 3, \ldots\}$ is countable basically by considering the ...
3 votes

Is a "non-analytic" proof of Dirichlet's theorem on primes known or possible?

I like the approach of looking for a sequence of integers with prime divisors that are guaranteed to be in a certain arithmetic progression. One can try taking the iterates of a polynomial, starting ...
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1 vote

Question on consecutive integers with similar prime factorizations

This is how Heath-Brown resolved the d(n)=d(n+1) problem ! The idea is due to Claudia Spiro who used it for d(n)=d(n+5040) but was unable to reduce the 5040. The same idea has now been used in many ...
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2 votes

Priming for the primes

(Very large) prime numbers are of particular importance in cryptography. See e.g. RSA algorithm. I remember also some quasi random number generators using prime numbers.
12 votes

Priming for the primes

Primes even appear outside mathematics, here is an example from biology: The fact that some species of cicadas appear every 7, 13, or 17 years and that these periods are prime numbers has been ...
5 votes

Priming for the primes

Regular polygons which you can construct with compass and straightedge have $2^k \cdot p_1 \cdot p_2 \dots p_n$ edges, where $p_i$ are all distinct Fermat primes (so not all primes). It's the reason ...
3 votes

Priming for the primes

There are maximal shift registers of length $\ p^k-1\ $ where $\ p\ $ is an arbitrary prime, and $\ k\ $ is an arbitrary natural number.
6 votes

Priming for the primes

Obvious, but still worth mentionning, because the trick is frequently helpful: to represent a collection of elements of different types, where you can have more than one element per type, and order ...
14 votes

Priming for the primes

The nonzero characteristics of fields are precisely the prime numbers.
12 votes

Priming for the primes

The abelian simple groups are precisely the groups of prime order.
11 votes
Accepted

Is Gauss's generalization of Wilson's theorem non-superficially related to the classification of moduli for which primitive roots exist?

I will show the two results are non-superficially related by showing one of them implies the other: the classification of moduli $n \geq 2$ for which the unit group $(\mathbf Z/(n))^\times$ is cyclic ...
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17 votes

Is Gauss's generalization of Wilson's theorem non-superficially related to the classification of moduli for which primitive roots exist?

Both Gauss' generalization, and the classification of moduli with primitive roots, are 'shadows' of the structural theory of the finite abelian group $G_m:=(\mathbb{Z}/m\mathbb{Z})^{\times}$. Gauss' ...
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2 votes

Twin Primes- Clement conjecture proof

It follows immediately (mechanically!) by $\rm\color{#90f}W$ = Wilson's theorem and (Easy) CRT as below, using $\!\bmod n\!+\!2\!:\ \color{#0a0}{(n\!+\!1)!} = \smash{\underbrace{(n\!+\!1)n}_{\large \ \...
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9 votes
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Divergence of primes dividing polynomials

Yes, the series diverges. We can reduce easily to the case of irreducible monic $Q$. Next, let $\alpha_Q(p)$ be the number of roots of $Q(x)$ in $\mathbb{Z}/p\mathbb{Z}$. Note that $M_Q$ is the set of ...
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2 votes

Primality test for numbers of the form $\frac{a^p-1}{a-1}$?

Following Max Alekseyev suggestions. The following Pari GP code provides false positives. You can play with limits $p$ and $a$ inside forprime and for loops respectively (last statement). ...
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