All Questions
Tagged with prime-factorization solution-verification
45
questions
11
votes
1
answer
393
views
What is wrong with this effort to generalize Bertrand's Postulate using the Inclusion-Exclusion Principle
The following argument is too elementary to be true but I am not clear where the error is. I would appreciate it if you could call out what is logically wrong or provide a counter example to one ...
- 10.5k
8
votes
5
answers
14k
views
Proving gcd($a,b$)lcm($a,b$) = $|ab|$
Let $a$ and $b$ be two integers. Prove that $$ dm = \left|ab\right| ,$$ where $d = \gcd\left(a,b\right)$ and $m = \operatorname{lcm}\left(a,b\right)$.
So I went about by saying that $a = p_1p_2......
- 2,148
6
votes
2
answers
205
views
(1) Sum of two factorials in two ways; (2) Value of $a^{2010}+a^{2010}+1$ given $a^4+a^3+a^2+a+1=0$.
Question $1$:
Does there exist an integer $z$ that can be written in two different ways as $z=x!+y!$,where $x,y\in \mathbb N$ and $x\leq y$?
Answer: $0!=1!$ so $0!+2!=3=1!+2!$
Question $2$:
If $...
- 3,569
6
votes
1
answer
1k
views
Strengthening the Sylvester-Schur Theorem
The Sylvester-Schur Theorem states that if $x > k$, then in the set of integers: $x, x+1, x+2, \dots, x+k-1$, there is at least $1$ number containing a prime divisor greater than $k$.
It has ...
- 10.5k
6
votes
1
answer
236
views
Is my proof that there are infinite primes actually valid?
I was trying to think of another way of showing that there are an infinite number of primes. I came up with the following argument, but I am not sure if it is valid. I don't know how to make it more ...
5
votes
3
answers
183
views
Sanity check on factorization of $\langle 5 \rangle$ in $\mathbb{Z}_{10}$
Looking at $\mathbb{Z}_{10}$, consisting of $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ with addition and multiplication adjusted so the ring is closed under both operations.
Clearly $5 = 3 \times 5 = 5 \times 5 =...
- 1,108
5
votes
0
answers
305
views
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
4
votes
3
answers
291
views
Is my intuition of "If $p \mid ab$ then $p \mid a$ or $p \mid b$" correct?
I'm studying number theory and I was given this Theorem to look at:
If $p \mid ab$ then $p \mid a$ or $p \mid b$
I had the following intuition for the problem or a proof of sorts if you will.
...
- 9,324
4
votes
2
answers
5k
views
If a prime $p$ divides $n^2$ then it also divides $n$ - Is this proof correct?
I'm learning Real Analysis by myself and I wanted to prove that if a prime $p$ divides $n^2$ where $n$ is an integer, then $p$ divides $n$ itself. I saw that proving this is the same as saying that ...
- 53
4
votes
2
answers
227
views
The Conjecture That There Is Always a Prime Between $n$ and $n+C\log^2n$
Let $n$ be 113. Use $n+C\log^2n$ to find the next consecutive prime or at least approximately how far away it is. Will you show me how to work this out step by step to show me how to use this formula? ...
- 849
4
votes
1
answer
56
views
Reasoning about a sequence of consecutive integers and factorials with hope of relating factorials to primorials
I am looking for someone to either point out a mistake or help me to improve the argument in terms of clarity, conciseness, and more standard mathematical argument.
Let $x$ be an integer such that $x,...
- 10.5k
3
votes
2
answers
754
views
How to calculate the number of possible multiset partitions into N disjoint sets?
I have made a Ruby program, which enumerates the possible multiset partitions, into a given number of disjoint sets (N), also called bins. The bins are indistinguishable. They can be sorted in any ...
- 155
3
votes
0
answers
92
views
Counting integers in a consecutive sequence where least prime factor is greater than $3$.
I am attempting to count the number of integers with a least prime factor greater than $3$ in a sequence of consecutive integers.
For example, if I count the number of integers in $10,11,12,13,14,15$,...
- 10.5k
2
votes
1
answer
260
views
Is it correct? Prove that any fraction can be reduced
I want to know if my prove is correct.
My goal is proving:
Hypothesis: $a,b \in \mathbb Z$ and $a,b \notin \{-1, 0, 1\}$.
Thesis: for all $ a, b$, exist $a',b'\in \mathbb Z$ that verify $$\...
- 1,277
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