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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.

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How to get this equation solved?

I came across this equation $$\sqrt a-\sqrt b=\sqrt 7-\sqrt 5$$ And you have to find the value of '$a$' and '$b$' when both of them are primes. The solution was $a=7, b=5$. Now, my question is, can't there be any other primes?
Rashed Tanvir
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Express the power of a natural number with the power of the product of prime factors

Given a natural number say $n \in \mathbb{N}$ with a prime factorization $p_1^{m_1} \cdot p_2^{m_2} \dots p_k^{m_k}$. If you take product of the prime factors $p_1 \cdot p_2 \dots p_k$ then the following holds: $$ \exists i,j \in \mathbb{N} ~~ n^i =…
Joachim
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Prime number of the form $A^{B^C}+D^{E^F}$

My Brother asked me what is the smallest prime number of the form $A^{B^C}+D^{E^F}$ where A,B,C are three distinct prime numbers, and D,E,F are 3 distinct primes that is Permutations of those 3 primes.I realize that we must arrange the exponents to…
Kenan D
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Which fields of math would I need to study to fully understand/solve the Riemann Hypothesis?

Apart from analytic number theory / complex analysis to actually knowing what it's about, which fields of math should I master to have a chance at solving it? I understand that the answer may come unexpectedly from anywhere, but I have limited time…
David Seelmann
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Given integers $q,n$, is there always an integer $x$ such $x^q - n$ is prime?

Does anybody know if this is true? I can't find references about it, also I can't prove to be true (or false). I think computing $x$ is a brute force task. Thanks (and sorry if I lost some basic concept).
Miguel Velilla
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$\limsup_{n\to\infty}\frac{g_n}{\log^3 p_n} < \infty$?

The following quote comes from Wikipedia http://en.wikipedia.org/wiki/Prime_gap "Usually the ratio of $g_n / \log p_n$ is called the ''merit'' of the gap $g_n$;. In 1931, E. Westzynthius proved that prime gaps grow more than logarithmically. That…
user238934
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prime numbered currency

The unit of currency is the Tao(t) the value of each coin is a prime number of Taos. The coin with the smallest value is 2T there are coins of every prime number Value under 50. Help! I don't under stand if the question means that 1coin equals…
Mag
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Something weaker than Goldbach's Conjecture

Inspired by this question on the "Sum of a odd prime and a odd semiprime as sum of two odd primes?", I wonder, if it is possible to show, that every even number $2n\ge 12$, can be written as a sum of a prime and semiprime $$p_1+p_2p_3=2n?$$
draks ...
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Is there a better way to search a string so that I don't get false positives?

I've been using this equation to find primes of a specified range, in this example the range is 10 to 10000. Go ahead and try it out, your monitor will not catch fire. The problem I'm now having is I'm getting false positives when the computer…
user24719
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Need help with formula for generating primes

I've done this: N[Sum[(10^(n*2) + 1)/(10^(n^2*2)*(10^(n*2) - 1)), {n, 1, Floor[49^(1/2)]}], (49)*2] 0.010202030204020403040206020404050206020604040208030404060208020604040409020404080208020606040210031 str = StringDrop[ToString@%,…
user24719
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Sets with prime subset sums

1. Given $M$ is it possible to pick a set of $T>M$ distinct numbers $a_i\in\Bbb Z$ such that sum of any $M+1$ or more of them will always be a prime and sum of any $M$ or less of them is always composite with no prime factors bigger that…
Turbo
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Is it true that in $2^n-1$, when $n$ is a prime number, you don't always get a Mersenne prime?

For $2^n-1$, where $n$ is a prime number, is it true that you don't always get a Mersenne prime? Remember, a Mersenne prime is a number that has a power of two subtracted by one and is then…
ReliableMathBoy
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Dividing an interval, such that the primes get divided even

I have an interval $[2,t]$ containing some number of primes. I now want to divide this interval into two intervals $a=[2,m]$ and $b=]m,t]$ such that the number of primes in $a$ and $b$ is almost the same. I am aware of the approximation…
utdiscant
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Properties of prime mod $3$

We know that if $p$ is a prime congruent to $3 \mod 4$, we cannot represent it as sum of two squares. Is there a positive property of such $p$? That is, do we have any statements that say "$p$ is a prime congruent to $3 \mod 4$ iff…
Turbo
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