Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [prime-factorization]

For questions about factoring elements of rings into primes, or about the specific case of factoring natural numbers into primes.

A natural number is prime if it has no positive divisors besides $1$ and itself. The fundamental theorem of arithmetic states that every natural number $n>1$ can be factored uniquely, up to a reordering of the factors, as a product of distinct prime numbers each raised to some power.

This concept holds in a more general setting though. A ring is called a unique factorization domain (UFD) if every non-unit element can be factored uniquely as a product of prime elements in the ring. The ring of integers is an example of a UFD.

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Are $14$ and $21$ the only "interesting" numbers?

The numbers $14$ and $21$ are quite interesting. The prime factorisation of $14$ is $2\cdot 7$ and the prime factorisation of $14+1$ is $3\cdot 5$. Note that $3$ is the prime after $2$ and $5$ is the prime before $7$. Similarly, the prime…
Simon Parker
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Do most numbers have exactly $3$ prime factors?

In this question I plotted the number of numbers with $n$ prime factors. It appears that the further out on the number line you go, the number of numbers with $3$ prime factors get ahead more and more. The charts show the number of numbers with…
SmallestUncomputableNumber
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Can I search for factors of $\ (11!)!+11!+1\ $ efficiently?

Is the number $$(11!)!+11!+1$$ a prime number ? I do not expect that a probable-prime-test is feasible, but if someone actually wants to let it run, this would of course be very nice. The main hope is to find a factor to show that the number is…
Peter
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Find a prime factor of $7999973$ without a calculator

How would you go about finding prime factors of a number like $7999973$? I have trivial knowledge about divisor-searching algorithms.
user523628
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Can a number be equal to the sum of the squares of its prime divisors?

If $$n=p_1^{a_1}\cdots p_k^{a_k},$$ then define $$f(n):=p_1^2+\cdots+p_k^2$$ So, $f(n)$ is the sum of the squares of the prime divisors of $n$. For which natural numbers $n\ge 2$ do we have $f(n)=n$ ? It is clear that $f(n)=n$ is true for the…
Peter
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Is there any other number that has similar properties as $21$?

It's my observation. Let $$n=p_1×p_2×p_3×\dots×p_r$$ where $p_i$ are prime factors and $f$ and $g$ are the functions $$f(n)=1+2+\dots+n$$ And $$g(n)=p_1+p_2+\dots+p_r$$ If we put $n=21$ then $$g(f(21))=g(231)=21.$$ I checked it upto $n=10000$, I did…
Pruthviraj
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A conjecture regarding prime numbers

For $n,m \geq 3$, define $ P_n = \{ p : p$ is a prime such that $ p\leq n$ and $ p \nmid n \}$ . For example : $P_3= \{ 2 \}$ $P_4= \{ 3 \}$ $P_5= \{ 2, 3 \}$, $P_6= \{ 5 \}$ and so on. Claim: $P_n \neq P_m$ for $m\neq n$. While working on…
Basanta Raj Pahari
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Have I discovered an analytic function allowing quick factorization?

So I have this apparently smooth, parametrized function: The function has a single parameter $ m $ and approaches infinity at every $x$ that divides $m$. It is then defined for real $x$ apart from a finite set of arguments. The animation above…
Pythagoras of Samos
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What are examples of irreducible but not prime elements?

I am looking for a ring element which is irreducible but not prime. So necessarily the ring can't be a PID. My idea was to consider $R=K[x,y]$ and $x+y\in R$. This is irreducible because in any product $x+y=fg$ only one factor, say f, can have a…
MyNameIs
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Numbers that are the sum of the squares of their prime factors

A number which is equal to the sum of the squares of its prime factors with multiplicity: $16=2^2+2^2+2^2+2^2$ $27=3^2+3^2+3^2$ Are these the only two such numbers to exist? There has to be an easy proof for this, but it seems to elude me. Thanks
barak manos
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Is the "cyclotomic diagonalization" always squarefree?

For every integer $n\ge 2$ , define $$f(n):=\Phi_n(n)$$ where $\Phi(n)$ is the $n$ th cyclotomic polynomial. This can be considered as the "cyclotomic diagonalization" Prove or disprove the conjecture that $f(n)$ is squarefree for every integer…
Peter
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How many primes do I need to check to confirm that an integer $L$, is prime?

I recently saw the 1998 horror movie "Cube", in which a character claims it is humanly impossible to determine, by hand without a computer, if large (in the movie 3-digit) integers are prime powers, i.e. they are divisible by exactly one prime…
ZKG
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Can it be proven/disproven that there are highly composite numbers that prime-factorize into larger primes such as $9999991$?

Of course, following the rules found by Ramanujan, such a highly composite number would need to factorize into all primes ascending up to 9999991 (with descending powers as the primes progress) so the highly composite number would be insanely…
Alexander Kalian
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Number of distinct prime factors, omega(n)

Is there a formula that can tell us how many distinct prime factors a number has? We have closed form solutions for the number of factors a number has and the sum of those factors but not the number of distinct prime factors. So for…
jessica
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Is $\frac{3^n+5^n+7^n}{15}$ only prime if $n$ is prime?

Let $f(n)=3^n+5^n+7^n$ It is easy to show that $\ 15\mid f(n)\ $ if and only if $\ n\ $ is odd. I searched for prime numbers of the form $g(n):=\frac{3^n+5^n+7^n}{15}$ with odd $n$ and found the following $n$ leading to a prime number :…
Peter
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