Questions tagged [semiprimes]

A semiprime is a natural number that is the product of two prime numbers. This tag is intended for questions about, related to, or involving semiprime numbers.

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Is it possible to generate a 1000 digit semiprime without knowing its two primes?

I studied this question. I'd like to modify it to make it more interesting and more challenging. Is it possible to generate a (very large, say, with 1000 digits) semiprime without knowing its two ...
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On Distribution of Semiprime Numbers. Done by Euler and Euclid?

This paper is very interesting. But, it amazes me that the work was done as late as year 2013! I thought that kind of work would have been done by somebody like Euler or Euclid. Since I am not a ...
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Can analyzing squre roots give useful insight into finding factors by Fermat factorization?

Given a semiprime: $N = 688307 = 431 * 1597$ $ceil(sqrt(N)) = 830$ $830^2 - N = 593$ $n^2 + 1660n − k^2 + 593 = 0$ $n = 184, k = 583$ $p1 = (830 + n) - k = 1014 - 583 = 431$ $p2 = (830 + n) + k = 1014 ...
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How can prime numbers be identified through characteristics of matrix/grid structures formed by polyhedra?

Have you come across a similar theory to predict prime numbers? I believe that I have identified a pattern by which prime numbers occur which is in the form: N = A(P,n) + B …where A is a function of a ...
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Can multiple pairs of primes produce the same semiprime?

What I mean by this is, say we have a semi prime number SP1 = P1 * P2, is it possible for SP1 to also be the product of some other pair of primes, say Px * PY? Or ...
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251 views

prime number (a form like Mersenne primes)

I found a form like Mersenne prime number and i wanted to be sure if its maybe better but i was wrong but still as good as Mersenne form its $(2^p+1)/3=P$ and p,P are primes P also can be a ...
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Prove this formula relating number of primes and semi-primes?

I think I have a messy proof which enables me to state for all $m > 2$ being a prime number: $$ \sum_{k=2}^{m-1} \pi(2k) - m+2 = \pi(m)+2S(m^2) $$ Where $\pi(x)$ is a function which counts the ...
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On variants of the Erdős–Moser equation as a reference request, and a computational problem as companion question

I'm curious to know if it is known some variants of the Erdős–Moser equation for different sequences of integers $a_m$ strictly increasing $1\leq a_1<a_2<\ldots<a_m<\ldots$ $$a_1^k+a_2^k+\...
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60 views

On an inequality that involves products and sums related to the sequence of semiprimes

A semiprime $s$ is a positive integer that is the product of two prime numbers, see Semiprine from the encyclopedia Wikipedia, thus corresponding to the sequence A001358 of the OEIS. I wondered if it ...
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Probability a natural number of the form $m^2 - n^2$ can be exactly factored as the product of $2$ primes?

Question Let $P$ be the probability that two integers where $m>1$is a fixed positive integer and $n$ is a randomly chosen such than $ m> n \geq 0 $? What is the probability $m^2 -n^2$ is ...
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Interesting patterns related to the sums of the remainders of integers

Let us define, $$r(b)=\sum_{k=1}^{\lfloor \frac{b-1}{2} \rfloor} (b \bmod{k})$$ After reading the posts Surprising fact about a certain number-theoretic function and Do primes have special sums... I ...
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All squarefree semiprime numbers that less than a certain number

How can i find all number $N$, where $N=p*q$. (Here, $p$ and $q$ are primes and $p<q$) ? For example: I want a formula that help me to find all $N<1000$
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Disprove that $ \int_0^1 \bigg((1-\sqrt{x})(1-\sqrt{1-x}\bigg)^n~dx $ can be written in the form $\frac{k\pi}{c}-\frac{P}{d}$

I want to disprove that for $n=2,3,4,\cdot \cdot\cdot$ $$ \int_0^1 \bigg((1-\sqrt{x})(1-\sqrt{1-x})\bigg)^n~dx $$ can be put into a form $\frac{k\pi}{c}-\frac{P}{d}$ where $k,c,d$ are integers and $...
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Extending Mertens' third theorem to composite numbers

Mertens' third theorem states that: $$\lim_{n \to \infty} \log{n} \prod_{p\le n} \frac{p-1}{p} = e^{-\gamma}$$ For $p$ a prime number. Is it possible to generalise that result when the product is ...
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Does the following function only generate primes and semi-prime* composites for all values of n?

*semi-prime, i.e. two distinct prime factors, but these could be raised to any powers. Generating function: $n^2 + 21n + 37$ First few values are:
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two semiprimes and a prime as three consecutive numbers

Three consecutive numbers comprise two semiprimes and a prime. Examples are 21, 22, 23 and 157, 158, 159. Do you think such trios are endless?
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Are Semiprimes in the Mersenne Sequence Bound (Eventually) to Occur at Terms of Prime Index?

Related to a question posed a year and a half ago on the site, Mersenne semiprimes I would now like to ask, that in the sequence of Mersenne numbers, does there exist a bound on the indices, say $K$,...
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Semiprime Factorization - Length against Time

I'm looking for current information on how long it takes to factor semiprimes of a given length. MIPS against bit length or decimal length of N using best known algorithm would be ideal but any ...
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Distribution of Pairs of Semiprimes

I have a means of generating pairs of semiprimes of a given bit length. The process entails Step 1. Generate a semiprime P1 of bitlength N, with factors close in length (close as defined in the ...
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largest known twin semiprimes and consecutive semiprimes

What are the largest known semiprimes differing by 2? What is the largest known pair of consecutive numbers that are both semiprimes? Has anyone computed this?
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How many primes and semiprimes are generated by this sum?

How many primes and semiprimes are generated in the numerator by this sum up to a given $k$? That is, after evaluating the sum and putting the fraction in reduced form, how often is the numerator ...
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Does the integral yield primes when the set of all $k$ consists of primes or semiprimes?

Let $\pi(x)$ be the prime counting function and $P$ be the set of primes. Is $\Psi(k)\in P$ when the set of all $k$ consists of primes or semiprimes? $$\Psi(k)=\int_0^k\pi(x)\pi(k-x)dx,$$ where $k\...
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$28222149$, a semiprime with amazing properties

The semiprime $28222149$ is a semiprime $S=A*B$ (with $A=3$ and $B=9407383$) such that -$A.B$ -$B.A$ -$S.A$ -$S.B$ -$A.S$ -$S.A.B$ -$S.B.A$ -$A.S.B$ -$A.B.S$ -$B.S.A$ -$B.A.S$ are all ...
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146 views

(Soft Question) Largest known semiprimes with no known factors

Is there a list, similar to prime numbers and probable primes, of the largest semiprimes with unknown factors? Is there a list of numbers that are either semiprime or prime, with no known factors? Is ...
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Might there be a Skewes number for semiprimes?

Briefly the question here is whether there is or could be a theorem analogous to that of Littlewood for semiprimes (generalized prime numbers which are products of two primes, repetitions allowed), ...
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808 views

Semiprime factorization

I was thinking about semiprime factorization, and I had an idea of an algorithm: Let's take a small semiprime for this example: 3053. So we have to primes ...
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Does the sequence $2n^2 +2n + 1$ consist only of primes and semiprimes? [closed]

The sequence is: $ 5, 13, 25, 41, 61, 85, 113, ... $ How do I prove that this sequence only contains primes and semiprimes. Would it be interesting if it did only contain primes and semiprimes?
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Always prime? $6n+1$ and/or $6n-1$, if neither divisible by $5$ nor semiprime

It seems that the sequence of all integers $N$ that are either $6n+1$ or $6n-1$ are all primes if $N$ is neither divisible by $5$ nor semiprime. It also seems that this sequence of primes includes ...
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Bounding the number of semiprimes of the form $2p\pm1$

I would like an upper bound on the number of semiprimes of the form $2p+1$ (and the same with $2p-1$), where $p$ is prime. Is there a general result I can apply? I have not studied sieve theory (but ...
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Mersenne semiprimes

Is it known does the set of Mersenne numbers contain an infinite number of semiprimes? By the same procedure as with primes if $n=abc$ with $a,b,c>1$ is composite number then $2^{abc}-1$ is ...
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135 views

primes and semiprimes found with $n=11,13,17,19 \mod 30$. [closed]

For $n=11,13,17,19 \mod 30$ what percentage of all primes and all semiprimes less than $100, 1000, 10000...$ will be produced by $n$? Up to $20,000$ I found $50\%$ were primes and more than $50\%$ ...
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between two semiprimes differing by 4 there are two primes

The two semiprimes 58 and 62 differ by four and have two primes 59 and 61 in the interval. Do you think this happens an endless number of times?
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Semiprimes with three (and more) numbers

Semiprimes (pq-numbers) guarantee that if p and q are prime numbers the only divisors of the result of p*q are p and q. My question is: Does this hold true for <...
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286 views

About square free semiprimes which are sums of two squares

This conjecture seems to be true: If the product of two different primes is a sum of two squares, the so are the primes. I've tested it for $pq<1000$, but would like to see a proof. Edit: If $p\...
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Could deciding whether a number is semiprime be easier than integer factorization?

Here http://primes.utm.edu/glossary/xpage/Semiprime.html I came across a number that baffles me! It has been proven(!) that the (constructed) number in the link is semiprime (the product of two ...
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Factoring semiprimes cost estimation

I have two problems that are the following. The first problem is the following: I need to estimate the cost of factorizing a given semiprime based on previous estimations. For example I have the time ...
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Concatenating the first n semiprimes (in order) to get a semiprime $469101415$...

The concatenation of the first $1,2,3,6,43$, and $61$ semiprimes (in order) is a semiprime (!), $4=2 . 2$ $46=2 . 23$ $469=7 . 67$ $469101415=5 . 93820283$ $4691014152122....121122123129$ (proven ...
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Has the Pascal-like triangle of semiprimes a hidden primality test or there is a counterexample?

This is an observation regarding the semiprimes, also named 2-almost primes, biprimes, or the product of two primes. Basically it seems that if the semiprimes are arranged in a Pascal-like ...
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Is always two times an even semiprime at a distance $1$ or prime to the closest previous odd semiprime?

This is an observation regarding the semiprimes, also named 2-almost primes, biprimes, or the product of two primes. This week I do not have a computer, only a tablet (hospitalized with a lot of free ...
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What is an upper bound for number of semiprimes less than $n$?

A semiprime is a number that is the product of two prime numbers. What is an upper bound for the number of numbers of the form $pq$ less than $n$? $p,q$ are prime numbers smaller than $n$.
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Can two different sets of semiprimes share the same sum?

Given: A semiprime number is defined as the product of two primes $a$, $b$, $c$, $d$, $e$ and $f$ are all distinct semiprimes Can a proof be constructed showing that the following equation cannot be ...
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Mersenne semiprimes with square indices

A Mersenne semiprime is a semiprime of the form $2^n-1$, it can be shown that $2^n-1$ can be a semiprime if and only if $n$ is either a prime or a square of a prime. There are plenty Mersenne ...
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Proof of the reciprocal of all semiprimes diverging?

$$\sum_{\text{semi-primes}}\frac{1}{s}=\frac{1}{4}+\frac{1}{6}+\frac{1}{9}+\frac{1}{10}\cdots$$ I almost positive that this sum diverges, but I would really like to see a very thorough proof. Thank ...
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Semiprime numbers which, along with their prime factors, generate many semiprimes by concatenation

There's something quite interesting about the number $1191$: this number is a semiprime ($1191= 3 \cdot 397$), the concatenation of its prime factors in any order are semiprimes ($3397$ and $3973$ ...
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Digital root of twin prime semiprimes

It appears that the product of any pair of twin primes (excluding the first pair 3 and 5) yields a semi prime whose digital root is equal to $8$. Example: $$ 17 \cdot 19 = 323 $$ The digital root of ...
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Pattern for generating primes and semiprimes?

First, is there a formula that can generate semiprimes in polynomial time? Also, I found this interesting pattern: $$3x+1, 3x+2$$ Inputting increasing natural x spits out $$7, 11, 13, 17, 19, 23, 25, ...
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Factorizable huge semiprime

I'm trying to understand how the number decimal The correct decimal number is: ...
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Extending the zeta function to semiprimes, etc.

The Riemann Zeta function is defined for $s > 1$ as \begin{align} &\prod _{n=1}^{\infty}\dfrac{1}{1 -\ p_{n}^{\ \ -s}}\\ \end{align} It is possible to extend the zeta function to semiprimes ...
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Fairly good semiprime estimate

I have found a nice estimate for the semiprime counting function \begin{align} &f_{2}(x):=x \log \left( \log (x)/\log \left( a+a/ \exp\left( (\log (\log (x)-2)-1)^2/2\right) (\log (x)-2) \right) \...
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Semiprime asymptotic step function

Since $$\pi_{(2)}(x)=\sum_{i=1}^{\pi(x^{1/2})}\left(\pi\left(\dfrac{x}{\text{p}_i}\right)-i+1\right),$$ where $\pi_{(2)}(x)$ denotes the semiprimes and $\text{P}_i$ is the $i$th prime, an asymptotic ...