-2
votes
1answer
14 views

Defining a function/(operator?) which will give a prime nearest to the product

Is there a function/(operator?) which gives a prime nearest to the product of two whole numbers like $x*y=p$ +-$d $ where $p$ is the nearest prime to the product and $d$ is the difference between the ...
1
vote
0answers
31 views

Schoenfeld's limits & almost primes

If I am correct in my understanding, the Tao-Green theorem employed almost primes as density normalisers. If $N_k(x)$ is the counting function of almost primes, is the study of almost primes 'useful' ...
11
votes
3answers
265 views

Why are conjectures about the primes so hard to prove?

I recently started learning number theory, and I've noticed there are many conjectures about the prime numbers that are unproven. Some examples would be whether there are infinite Mersenne, ...
1
vote
1answer
98 views

Analogue of prime numbers in addition? [closed]

What is the analogue of prime numbers in addition?
4
votes
4answers
220 views

By definition, how is a prime number represented?

Even numbers can be easily represented as $2n$. Odd numbers as $2n+1$. An exactly divisible operation can be defined as $n = dq$. But, is there an specific way of representing a prime number, ...
5
votes
0answers
261 views

Logical consequence of Euclid's theorem

Are there any far reaching non-trivial consequences of Euclid's infinitude of primes where theorems make use of it? Wikipedia does not have the list of applications of this theorem, rather modern ...
6
votes
3answers
176 views

Currently, what is the largest publicly known prime number such that all prime numbers less than it are known?

So recently, an absurdly large prime number was found, but a lot of prime numbers less than it are still not known. I am wondering up to where we know all the primes. I put "currently publicly known" ...
9
votes
5answers
2k views

Why do we consider prime numbers important, and what are their applications other than number theory in pure math?

Why do we consider prime numbers important, and what are their applications other than number theory in pure math? I know that Number theory is devoted to studying prime numbers, but there must be ...
6
votes
3answers
348 views

Characterizations of primes

Let $\mathbb{P}$ be the primes set. We know from Wilson's Theorem that $$(p-1)!\equiv-1 \pmod p \iff p \in \mathbb{P}$$ What another formulas we have with an if and only if ($\iff$) statement to ...
10
votes
1answer
313 views

What is the intuitive meaning of “conspiracy” in number theory?

Assuming very little number-theoretic background from my part, could you please explain me what is the intuitive meaning behind "conspiracy" in number theory? There is no formal entry on Wikipedia and ...
10
votes
5answers
243 views

Is there a single or best reason that 2 is an exceptional prime?

I've recently been studying some elementary number theory, and I've frequently come across the fact that there are a fair number of results (the main one being the law of quadratic reciprocity) for ...
43
votes
4answers
2k views

How to understand and appreciate the prime number industry?

Why would I want to buy prime numbers? There is a website (found it!) selling a table of 400 digit primes for twenty dollars. Like an updated version of this. I have a layman's idea that prime numbers ...
4
votes
1answer
240 views

Are there other pseudo-random distributions like the prime-numbers?

Does there exists other structures in math, which are seemingly random, but deterministic, and follow rules similar to the prime numbers, by rules I mean there must be statements similar to goldbach's ...
6
votes
2answers
298 views

A Question on RH relating to Prime Number theorem

Well, in a previous post regarding the explanation of Riemann Hypothesis Matt answered that: The prime number theorem states that the number of primes less than or equal to $x$ is approximately ...