Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [prime-numbers]

Prime numbers are natural numbers greater than 1 not divisible by any smaller number other than 1. This tag is intended for questions about, related to, or involving prime numbers.

A prime number (or a prime) is an element of the greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number ... The fundamental theorem of arithmetic establishes the central role of primes in :

Any integer greater than 1 can be expressed as a product of primes that is unique up to ordering.

Here you get the first 50 millions of primes.

The concept of prime numbers is extended in ring theory, where an element $p$ of a ring $R$ is prime if and only if whenever $p\mid ab$, then $p\mid a$ or $p\mid b$.

One can easily see that this extends the definition of prime numbers in the natural numbers.

12562 questions
2
votes
0 answers

Does this formula always yield a prime?

Somehow $\tau=1.2516475977905$ appears to have the property that $$ \left\lfloor 2^{2^{{\,}^{\cdot^{\cdot^{\cdot^{\tau}}}}}}\right\rfloor $$ is always a prime. Here $\lfloor x\rfloor$ denotes the floor function that returns the integral part of…
String
  • 18,395
2
votes
0 answers

Proof without using the proof of contradiction

By using the proof by contradiction I can determine that the root of a prime number is irrational. But how can I proof this by using the rational roots test to find rational factors of $x^n - p$. How does this information leads me to get a…
user164612
2
votes
2 answers

If $n>1$ has $r$ different prime factors, then the totient is bounded by $\varphi(n) \geqslant n/2^r$?

I want to prove that if $n>1$ has $r$ different prime factors, then $$\varphi(n) \geqslant \frac{n}{2^r}.$$
Paulo Vandeveld
  • 89
2
votes
3 answers

Quadratic that yields the longest prime sequence?

The quadratic $n^2+n+41$ yields prime numbers all the way up to $n=40$ before it fails (pretty cool!). My question is: Do you know of a quadratic that can 'last even longer'?
Trogdor
  • 10,331
1
vote
1 answer

Consecutive prime numbers

Let's assume $k$ and $n$ are consecutive prime numbers, $k \lt n$. An axiom: for any such $k$ and $n$, $k^2 \gt n$. This seems "obviously" true to me, but could you please prove me wrong? Or if it is correct, could you please help me prove it?
kkodev
  • 119
1
vote
1 answer

can Sophie Germain prime be arbitrarily many?

We know that there exists arbitrarily long prime arithmetic progressions by BEN-TAO. Together with Dirichlet's theorem on arithmetic progressions， can we address that Sophie Germain prime number be arbitrarily many? Note that the arithmetic…
Ocean Yu
  • 139
1
vote
3 answers

If $m$ and $n$ are integers with $\gcd(m,n) = 1$, prove that $\sigma(mn)= \sigma(m)\sigma(n)$.

If $m$ and $n$ are integers with $\gcd(m,n) = 1$, prove that $\sigma(mn)= \sigma(m)\sigma(n)$. I am thinking about using the formula for $\sigma(p^k)$ where $p$ is prime. It follows from the geometric series that $$\sigma(p^k) = 1 + p + p^2…
Pasie15
  • 491
1
vote
2 answers

Can we find an integer $m$ such that: $2^{2p-2}-2^{p}+3=m²$

Let $p$ a prime number. Can we find an integer $m$ such that: $$2^{2p-2}-2^{p}+3=m²$$
DER
  • 3,011
1
vote
0 answers

Questions about primes made from consecutive numbers starting from 1

Similar to: Does there exist a prime that is only consecutive digits starting from 1? Let $b_n=\overline{a_1a_2a_3\dots a_n}$ and $a_n=n$. For example $b_{11}= 1234567891011$. I have a couple of questions about the primes in the sequence $b_n$.…
Joao
  • 1,390
1
vote
2 answers

Can anyone prove this formula?

I found formula below$$p_n=6\left \lfloor \frac{p_n}{6}+\frac{1}{2} \right \rfloor+\left ( -1 \right )^\left \lfloor \frac{p_n}{3} \right \rfloor$$ for $n>2$, $p_n$ is prime number sequence. Can anyone prove this formula?
chimpanzee
  • 157
  • 1
  • 6
1
vote
1 answer

distribution of elements related to prime numbers

here is my question. Fix any prime $p$ and consider the set of all elements of the form $\frac q{p^k}$, where $q$ is any other prime and $k$ is the unique integer such that $\frac q{p^k}$ belongs to the interval $]1,p[$. Is it true that this set is…
Pikapika
  • 13
1
vote
3 answers

The Largest Prime Less Than the Square of a Prime

The first prime is two. Two squared is four. The largest prime that is less than four is three. The set of primes is 2,3,5,7,11,13,17,19,23,29,31... The set of their squares is 4,9,25,49,121,169,289,361,529,841,961... The set of numbers which I am…
Jeffrey Young
  • 841
1
vote
1 answer

Reducing fractions with prime number denominators into additions of unities.

So I'm working on practicing reducing fractions into additions of unities (like ancient greek math). It's actually very enjoyable, except when I end up running into a fraction with a prime number as the denominator, as I rely on the multiples of the…
Phi pi hohum
  • 13
1
vote
2 answers

Dividing a product of different primes by another prime

A relatively straight forward question. If I were to multiple any amount of different prime numbers together say 7*3*11, is it possible to divide the product by a single other prime number say 23 and have it result in an integer? So in essence, will…
Shiri
  • 111
1
vote
0 answers

RSA aloghorithm - stuck on d

I'm sorry in advance if this sort of question has been posted before. I couldn't find it. I'm clearly an idiot, and I clearly need help, so here I am. I have a homework assignment which overall is fairly simple but I can't see where I'm going…
user2994883
  • 11
1 2 3
…
60 61