All Questions
20
questions
5
votes
0
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305
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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 ...
- 1,405
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
1
vote
0
answers
31
views
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
0
votes
0
answers
214
views
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\...
- 1,282
-3
votes
1
answer
332
views
A Simple Proof of the FTA using only elementary theory?
UPDATE/WARNING: DO NOT READ (WASTE OF TIME)
The effort I put in here is now an embarrassment as it goes nowhere. I can't delete this posting since there is an answer, but if a moderator could delete ...
- 11.5k
1
vote
2
answers
171
views
Application of Unique Factorisation Theorem in Proof
CONTEXT: Proof question made up by uni math lecturer
Suppose you have $x+y=2z$ (where $x$ and $y$ are consecutive odd primes) for some integer $z>1$, and that you need to prove that $x+y$ has at ...
- 359
1
vote
1
answer
146
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Integers of this form that pass the Fermat Primality test are prime, proof?
If an integer, $2p + 1$, where $p$ is a prime number, is a divisor of the Mersenne number $2^p - 1$, then $2p + 1$ is a prime number.
My argument is that because divisors of the Mersenne number $2^p -...
- 13
0
votes
1
answer
151
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Proof verification: finding all prime numbers in the form of $n^3-1, n>1$
Let $p$ be a prime number of the form $p = n ^3 - 1$ for a positive integer $n \geq 2$.
Then, factoring the difference of perfect cubes, we obtain $p = (n-1)(n^2 + n + 1)$.
Since $p = 1 \cdot p$ as ...
- 1,271
11
votes
1
answer
393
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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
1
vote
0
answers
108
views
Counting $x$ where $an < x \le (an+n)$ and lpf($x$) $ \ge \frac{n}{4}$ and $1 \le a \le n$
Let lpf($x)$ be the least prime factor of $x$.
It seems to me that if:
$1 \le a \le n$
$n \ge 128$
$an < x \le (an+n)$
lpf($x$) $\ge \frac{n}{4}$
Then, for all $y$ where:
$an < y \le (an+...
- 10.5k
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
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 ...
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
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