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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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For a given number k，the existence of prime p > k that satisfied 10, 100,...,10^(p-1) are complete residue system modulo p

Is there a way to prove or disprove： For any given number $k$，there must be prime $p$ that satisfied $10, 100,\dots,10^{p-1}$ are complete residue system modulo $p$. for example： If k = 6, I can choose p = 7 or…
Ichungchen
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Does the number of largest sub-primes required to compute the sum within a prime number ever exceed 3?

Let's say for a prime number $P$, I compute the sum involving $P' + P''$, where $P'$ is the largest sub-prime number below $P$ and $P''= P - P'$, such that $P''$ is the largest sub-prime below P' to fit in this sum exactly. For example: $97 = 89 + 5…
Kaustubh Sinha
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Approximate sum of prime numbers

An approximate sum of prime numbers smaller than a given number 'n' can be found by the following formula $$\frac{n^{2}}{4\log_{2.75+\frac{0.325}{\log_{5}(\sqrt{n})}}(\sqrt{n})}$$ for example if n=1000,my formula will give value 77425, or another…
Srbin
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Proofs with prime numbers.

Show that if $ n ^ 2 + 2 $ is prime, then $ 3 \mid n $. Show that the only prime of the form $n ^ 3-1$ is $7$. For the 2. Let $ p = n ^ 3-1 $. Let's see that the only possibility for $ p = 7.$ For that note that $ n ^ 3-1 $ can be decomposed as…
asd asd
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Term for number only divisible by 1, itself, and its square root

Do numbers only divisible by 1, themselves, and their square roots have a specific term? Seems like they're all squares of primes: 1, 9, 25, 49, 121, 169... I feel like I'm missing something really obvious, I apologize if that's the case. Searching…
primeroot
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How to solve large numbers if they are a prime number? Eg Is 3599 a prime number?

I am reading the book, the Flash Boys, and in it is a trick interview question as per the title. The person solves the problem so: 3599 = (3600 – 1) = (60² – 1²) = (60 – 1) (60 + 1) = 59 × 61 3599 = 59 × 61 not a prime number. I am curious about…
giorgio79
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Mathematical way of proving an equation does not produce a prime number for all numbers in a set?

Given $k = n^2 + 9n + 1$ Prove that the statement “$k$ will be a prime number for all integer values of $n$ from $1$ to $9$”, is wrong. You can try the numbers $1$ to $9$ in turn, and determine that $6$ proves the statement wrong. But is there an…
Harry B
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Explain the difficulty of find primes with the same last digit

Refer to https://oeis.org/A340800 to notice that the number of primes between two primes having the same last digit is increasing as the primes themselves increase. Is there an explanation for this? How can the size of primes have any influence on…
J. M. Bergot
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Find all $x$ such that $|4x^2 - 12x - 27|$ is prime

Find all integers $x$ such that $|4x^2 - 12x - 27|$ is prime. I first factored $|4x^2 - 12x - 27|$ as $(2x+3)(2x-9),$ but I was unsure where to go from there.
questionasker
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Can we deduce that set of $a,b,c$ is the whole set of integers?

We say that three positive integers $x,y,z$ are coprime if there exist three integers $a,b,c$ such that $$ax+by+cz=1$$ if we run $x,y,z$ over all possible coprime positive integers. Can we deduce that set of $a,b,c$ is the whole set of integers?
Safwane
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Show that $-(\lambda_2\lambda_4-\lambda_2\lambda_6-\lambda_4\lambda_5)/(\lambda_1\lambda_4\lambda_5+\lambda_2\lambda_3\lambda_6)=A$

Let $A,B,C$ be three pairiwise coprime positive integers, i.e., there exist six integers $\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4},\lambda_{5},\lambda_{6}$ such…
Safwane
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How one can guaranty that $λ₁λ₄λ₅+λ₂λ₃λ₆$ is not a zero

Let $A,B,C$ be three pairiwise coprime positive integers, i.e., there exist six integers $λ_{1},λ_{2},λ_{3},λ_{4},λ_{5},λ_{6}$ such that $$λ_{1}A+λ_{2}B=1$$ $$λ_{3}A+λ_{4}C=1$$ $$λ_{5}B+λ_{6}C=1$$ Solving with respect to $A,B,C$, we…
Safwane
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I am asking if it is possible to choose the following case: $λ_{2}<λ_{5}$ and $λ_{2}λ_{6}>0$

Let $A,B,C$ three coprime positive integers, i.e., there exist six integers $λ_{1},λ_{2},λ_{3},λ_{4},λ_{5},λ_{6}$ such that $$λ_{1}A+λ_{2}B=1$$ $$λ_{3}A+λ_{4}C=1$$ $$λ_{5}B+λ_{6}C=1$$ I know that by the Bézout theorem…
Safwane
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I am stuck with some prime number problems. Are there any hints you could give me?

Find all natural values of n which make the value of (n^3 - 1)/5 a prime number. Find all values of p which make p, p + 2 and p + 4 prime numbers. Find all prime numbers p which also make p + 10 and p + 20 prime too. Thanks in advance!
Ognjen Ognjanovic
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Prove that each prime divisor of $(2^p)-1$, where $p$ is a prime, is greater than $p$.

How would I prove that if $(2^p)-1$ is not prime, there is not a prime number less than p that divides it? (any hints for a proof by contradiction?) I have assumed that there is a prime divisor less than p, let's say q. But I am unsure of where to…
Isabelle Stepler
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