All Questions
Tagged with prime-factorization solution-verification
45
questions
5
votes
0
answers
300
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Valid Elementary Proof of the Bertrand-Chebyshev Theorem/Bertrand's Postulate? [closed]
$\textbf{Theorem}$ (Bertrand-Chebyshev theorem/Bertrand's postulate): For all integers $n\geq 2$, there exists an odd prime number $p\geq 3$ satisfying $n<p<2n$.
$\textit{Proof }$: For $n=2$, we ...
- 445
0
votes
0
answers
87
views
I am trying to find the maximum gap between any prime and the nearest prime (whether smaller or bigger)?
I am trying to find the maximum gap between any prime and the nearest prime number (whether smaller or bigger)? Here is what I have:
Assuming: I don’t know whether any of the multiples that are ...
- 1,415
1
vote
1
answer
113
views
Elements of the sequence have a prime factors an element of the sequence
I am reading the following problem:
For the sequence $T=3, 7, 11, 15, 19, 23, 27 ...$ prove that every
number in $T$ has a prime factor that is also in $T$
My approach:
The sequence is of the form $...
- 1,609
0
votes
1
answer
98
views
Show No. of trailing zeros in base $p$ representation = power of prime $p$. .
Let $N$ be an integer $> 1$, and $p$ a prime, then need show:
No. of trailing zeros in base $p$ representation = power of prime $p$ in prime factorization.
Say, $N= 31500 = 2002000_5$, and $31500= ...
- 4,534
1
vote
1
answer
25
views
Can you just assume without proof for coprime bases?
Let $x,y ∈ \mathbb{Z}$ and $a,b ∈ \mathbb{Z+}$, where $a\ne b$ and a,b are coprime
Let $a^x=b^y$
The only solution is when $x = 0$ and $y = 0$, but is it necessary to prove it?
I've got as far as ...
- 113
1
vote
0
answers
142
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How many divisors does an integer with prime factorization have
Let $n$ be an integer with prime factorization $n = p_1^{n_1} p_2^{n_2} ... p_k^{n_k}$ , where the $p_i$ are distinct primes. How many positive divisors does the integer $n$ have?
Here is what I have:
...
- 303
2
votes
0
answers
37
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Prime inert in each subfields never occurs
Suppose you are working with $K=\mathbb{Q}[\sqrt{m},\sqrt{n}]$ and $K_1=\mathbb{Q}[\sqrt{m}],K_2=\mathbb{Q}[\sqrt{n}], K_3=\mathbb{Q}[\sqrt{k}]$ where $m,n$ are squarefree integers and $k = mn/gcd(m,n)...
- 1,078
1
vote
0
answers
63
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Prove or disprove that $\frac{(x+n)!}{x!}$ is not divisible by $n$ distinct primes where each prime is greater than $\frac{(x+n)e}{n}$
Is it true that for integers $n > e, x \ge n$, it is impossible for $\dfrac{(x+n)!}{x!}$ to be divisible by $n$ distinct primes where each prime is greater than $\dfrac{(x+n)e}{n}$?
Example:
$x=n=...
- 10.5k
1
vote
1
answer
135
views
Prove that, for every natural number, their factorization as primes is unique
I need some feedback on this proof I wrote that:
$$\forall n\in\mathbb{N} \text{ assumed the existence of a factorization of } n \text{ as } n = p_1p_2\cdots p_k, \text{ where } p_i, (1 \leq i \leq k)...
- 701
1
vote
0
answers
31
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Does it follow that there are at most $n+\pi(n-1)$ distinct primes that divide $\dfrac{(x+n)!}{x!}$
I've been thinking about the total number of distinct primes and it occurred to me that for any integers $x > 0, n > 0$, there are at most $n+\pi(n-1)$ distinct primes that divide $\dfrac{(x+n)!}...
- 10.5k
1
vote
0
answers
50
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Finding the number of divisors that satisfy some condition - What am I doing wrong?
Let $N$ = $2^1$$^5%$$3^1$$^6$. How many divisors of $N^2$ less than $N$ are there which don't divide $N$
MY ATTEMPT:
We have:
$N^2$ = $2^3$$^0%$$3^3$$^2$. The number of divisors of $N^2$ which do ...
- 902
2
votes
1
answer
639
views
Prime elements in the subring $R = \mathbb Z + x \mathbb Q[x]$ of $ \mathbb Q[x]$
Two questions I was given and want to make sure my reasoning is correct.
(1) Show that if $p$ is prime in $Z$, then $p$ is prime in $R = \mathbb Z + x \mathbb Q[x]$ (the subring of polynomials ...
- 151
0
votes
0
answers
214
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Need help interpreting this formula for the number of Goldbach partitions
1: Formula for the number of Goldbach partitions.
Let $g\left(n\right)$ denote the number of Goldbach partitions of even integer $2n$:
$$g_{\left(n\right)}=\sum_{3\leq p\leq2n-3}\left[\pi\left(2n-p\...
- 1,284
1
vote
1
answer
169
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Degree $3$ polynomial with constant coefficient $2010$
$\mathbf{Statement}$: Let $P$ be a degree $3$ polynomial with complex coefficients such that the constant term is $2010$. Then $P$ has a root $\alpha$ with $|\alpha|>10$. (TRUE OR FALSE?).
...
- 2,620
2
votes
2
answers
150
views
A proof of the existence of prime numbers right after 'booting-up' the counting numbers?
Not sure if the following argument is circular in nature or breaks down in some other way.
We've defined the counting numbers $n \ge 1$ with the two familiar binary operations of addition and ...
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