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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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Proving finite vs infinite representation of $p/q$ in base-$b$?

Reading up on positional notation and converting between different bases, I came across this statement: For integers p and q with gcd(p, q) = 1, the fraction p/q has a finite representation in base b if and only if each prime factor of q is also a…
ivan
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Question on Identifying Prime Factors of a Very Large Number

Let $P$ be the product of all numbers less than $90$. Find the largest integer $N$ so that for each $n∈$ {$2,3,4,...,N$}, the number $P+n$ has a prime factor less than 90. Upon first thinking about this question, my plan of action was to find the…
SolvingTrainee
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pairs of prime numbers, $x$ and $y$, where $x=(y \times c)+1$

I need to find pairs of primes that fit the equation that $x=(y \times c)+1$, where $c$ is a positif integer. or you can say that $x\equiv 1\pmod y$ is there a criterion in finding these prime pairs, or is it just coincidences. if so then what are…
adli farhan
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How one can get an estimate or a range for the integer $n$

Let $(p_{n})_{n≥1}$ be the sequence of prime numbers. I have an inequality of the form: $$p_{n}
Safwane
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Does $n = 2p$ where $p$ is prime, have fewer prime pairs than $n \neq 2p?$

Does an even integer $n = 2p$ where $p$ is prime, have relatively fewer prime pairs $(p_1, p_2)$ such that $p_1 + p_2 = n$, than an even integer $n \neq 2p$?
temp watts
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Finding bound counting primes which end by $3$

I am trying to count the number of prime numbers which end by digit "$3$" such as $3, 13, 23$, etc. and are below $10^6$. The number of primes existing below $10^6$ is known empirically to be $78~ 498$. Because, at most 1 every 10 numbers ends by…
Jose
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largest prime remainder from three consecutive primes

From three consecutive primes 89 results from $223*211mod199$=89. Is this the largest prime remainder? For p(n)=then nth prime, the form is p(n+1)*p(n+2)modp(n)=prime.
J. M. Bergot
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Residue Classes

I am trying to show that: $$\sum\limits_{\beta \in \mathbb{Z}_p^*}{\beta^{-1}}=\sum\limits_{\beta \in \mathbb{Z}_p^*}{\beta}=0$$ Where p is an odd prime. I really dont know where to start, but my best guess is that because B and the inverse of B…
Steven
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what is the property of this numbers

A multiple of $3$ gives $0,3,6,9,12,15,18,21,24,27$ which rearrange the last bit of each number gives $0,1,2,3,4,5,6,7,8,9$ once. The same can be said of $7$ and $9$ as well which gives $0,7,14,21,28,35,42,49,56,63$ and…
user1655072
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Partial part of fermat's little theorem

Am I fool? I cannot understanding the partial part of the proof. $\{1,2,...,p-1\}\equiv \{a,2a,...,(p-1)a\}\ (mod\ p)$ Why above statement is true? How I remove a??
baeharam
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Can prime numbers be defined formally like this

We can say that $p$ is prime if and only if : $$p\ne 1$$ $$p\space is\space prime \leftrightarrow \left\{(1|p)\wedge(p|p)\wedge\forall i\space2\leq i\leq p-1 (p\equiv r(mod\space i),r\neq0\right),(p,r)\in N\}$$ $$Edit: $$ $$p\ne 1$$ $$p\space…
Noodle
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Last digit of a triangular number is the midpoint between two primes

If a triangular number is even, add and subtract 1 to see if you have found two primes, as in 6 +/1 gives 5 and 7. If the triangular number is odd, add and subtract 2 to see whether or not both are primes, as in 105 +/-2 gives 103 and 107. A small…
J. M. Bergot
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Construct relatively prime pairs from set of nonrelatively prime numbers

Is it possible? Given a set of numbers S such that every possible combination of the numbers in S are nonrelatively prime, can you construct a pair of relatively prime numbers by adding up arbitrary combinations of the numbers from S?
Jin
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Find the three digit prime number.

What is the largest three-digit prime each of whose digits is a prime? - I believe it is 773, but correct me if I'm wrong.
A.Rin
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Integer linear combinations of three prime numbers

I'm looking for four prime numbers $a < b < c < z$, so that $\forall p, q, r, s \in N_{\geq 0}$: $pa+qb+rc = sz \rightarrow$ "All of $p$, $q$ and $r$ are multiples of $z$" Do such prime numbers $a, b, c, z$ exist? For example, $2, 5, 7, 11$…
Albert Hendriks
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