# Tag Info

### 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 ...
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. ...

### 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 ...

### 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.

### 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 $N$...

### 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 ...

### 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 ...

### 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 ...

### 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 ...
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 ...

### 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.

### 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 ...

### 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 ...

### 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.

### 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 ...

### Priming for the primes

The nonzero characteristics of fields are precisely the prime numbers.

### Priming for the primes

The abelian simple groups are precisely the groups of prime order.
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 ...
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' ...