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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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proof of the prime number formula(It's real)

Proof that this formula always generates primes.Also it generates all primes(grate prime numbers formula) I have tried this formula it also generates primes orderd except the prime 2. I am really interested in this formula because it generates all…
Taha Akbari
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How many ordered triples of primes $(a, b, c)$ exist

Is there any easy way to find the answer to the following problem without trying numbers one by one? How many ordered triples of primes $(a, b, c)$ exist such that $a+b+c=37$?
learning
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Is there any formula to find prime numbers

I have found from a site this formula: Ok.I have found that this formula is correct.see the reason below. This part of formula is always $1$ or $zero$. it's zero when $(2m)!+1$ isn't dividable to $2m+1$. And when it's zero this formula will give…
Taha Akbari
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divisibility of a product of (unique) primes by another product of (unique) primes

Assume that A is a set of prime numbers (without duplicates). Let P_A be the product of all elements of A. Assume that B is another set of prime numbers (again without duplicates) and let P_B be their corresponding product. If P_A divides P_B, that…
just-a-programmer
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Prime numbers sum (interchanged digits)

Here is a question which is really troublesome: Let N be a 2 digit prime number. When the digits are interchanged we get another prime number M. If M + N =176, find N-M.
crimsonKnight
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A Mersenne prime has $17 425 170$ digits. How many digits need to be checked to know that this is a prime?

A Mersenne prime has $17 425 170$ digits. How many digits needs to be checked to know that this is a prime? I know that the square rot of a number digit needs to be checked to know if it is a prime, but I have no idea what to do here.
Apple Pie
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Prove that 2003 is prime.

How do I prove that 2003 is a prime number? It's not very impressive going through factors and searching the internet only gives answers to smaller numbers. Can anyone help?
Luke
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Show that for any integer $n\geq 1$, all the numbers $(3 n + 1)^5 + 5$ are composite (i.e. not prime).

I would appreciate if somebody could help me with the following problem: Q: Show that for any integer $n\geq 1$, all the numbers $(3 n + 1)^5 + 5$ are composite (i.e. not prime). I expand the formula $$(3 n + 1)^5 + 5=243 n^5+405 n^4+270 n^3+90…
Young
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Detecting if a decimal is terminal or not?

If $x$ is prime and $1000000000000 < x < 1000100000000$, how can I detect if $1 / x$ will result in a terminal decimal or not? I can use programming.
SuprDewd
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If $p$ is prime and $p$ divides $a^2+b^2 $and $p$ divides $c^2+d^2$ does $p$ divide $a^2-c^2$?

I need some help getting started on proving this. If $p$ is prime and $p$ divides $a^2+b^2$ and $p$ divides $c^2+d^2$ does $p$ divide $a^2-c^2$? Thanks in advanced!
user317481
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Find three integers x, y, z such that there is a prime number that divides x and y, y and z, but not x and z?

Recall that 1 isn't prime or composite. I cannot think of any combinations. I am trying to find a counterexample to show that this relation is not transitive.
Arctix
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How to generate a list of primes using the fundamental theorem of arithmetic

I've been told it's possible to generate a list of primes using the fundamental theorem of arithmetic, will someone show me how?
Leila
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Prove that 2016 cannot be expressed as sum of prime and triangular number

As in the title. I've read that 2016 cannot be expressed in such form, but I've completely no idea, how could this fact be proven.
user263286
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Can $a(2as-4a-s-4)$ be a prime, except when $a=1$

Can $a(2as-4a-s-4)$ be a prime, except when $a=1$. And both $a$ and $s$ are positive integers
redelectrons
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Can there be a prime of the form $2y^2s + ys - 4y^2 +1$

Can there be a prime of the form $2y^2s + ys - 4y^2 +1$ where $y$ and $s$ are positive integers Forgot to say, $s \ge 3$ and $y \ge 1$
redelectrons
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