Questions tagged [erdos-conjecture]
10 questions
6
votes
1 answer
$a_n$ increasing, $\sum\frac{1}{a_n}$ diverges $\implies\exists\ n_1
I want to know if the following proposition is true or false, or if it is more-or-less equivalent to Erdos conjecture.
Proposition: For any fixed $\varepsilon>0$ and any increasing real sequence
$(a_n)_{n\in\mathbb{N}}$ such that…
Adam Rubinson
- 24,300
4
votes
0 answers
An Exponential Diophantine Equation related to the Erdos Ternary Conjecture
I am studying my conjecture that $2^n$ and $2^{n+1}$ have a common digit in base 5 if $n>6$. I believe that this conjecture is true provided that
$$
2^x=12(5^{y_1}+ \dots +5^{y_k}) + 5^z - 1
$$
has no solutions in integers where $x \ge 0$, $0\le…
Lewis Baxter
- 147
- 5
4
votes
0 answers
Let $A_N$ be the subset of $\{1,\ldots,\ N\}$ that has no $3$-term A.P's and maximises $\sum_{n\in A_N}\frac{1}{n}.$ Does $A_N\to A003278?$
This question is about A003278, which we define as: $\ a(1) = 1,\ a(2) = 2,\ $ and thereafter $\ a(n)\ $ is smallest number $\ k\ $ which avoids any $\ 3-$term arithmetic progression in $\ a(1),\ a(2),\ \ldots,\ a(n-1),\ k.$ The first few terms…
Adam Rubinson
- 24,300
2
votes
1 answer
On a weak polynomial version of Erdős conjecture
This question is motivated by Erdős conjecture on arithmetic progressions. It is a weaker conjecture than Erdős' one.
Erdős conjecture on arithmetic progressions states, "If $A$ is a large set, then $A$ contains (nontrivial) arithmetic progressions…
Adam Rubinson
- 24,300
2
votes
1 answer
For every large set $A\subset \mathbb{N},$ there is a concave subsequence of $A$ of length $k$ for every $k\in\mathbb{N}$.
Proposition: Suppose $A\subset \mathbb{N}$ is a large set in the sense
that
$$ \sum_{n\in A} \frac{1}{n} = \infty.$$
Then there exists $a_1 < a_2 < \ldots < a_k,\ $ (not necessarily
consecutive) members of $A,$ such that
$$ a_k - a_{k-1} \leq…
Adam Rubinson
- 24,300
2
votes
1 answer
Has this weak version of Erdős Conjecture on arithmetic progressions been proven, or is it still an open problem?
This question is motivated by Erdős conjecture on arithmetic progressions. It is a weaker version of Erdős Conjecture, but I do not know how to prove it.
Erdős conjecture on arithmetic progressions states that, "If $A$ is a large set, then $A$…
Adam Rubinson
- 24,300
2
votes
1 answer
How to show $A_x:= \{ \left\lfloor x^n\right\rfloor:n\in\mathbb{N}\}$ is **not** an additive basis (of order $k=2$) of $\mathbb{N}$ if $x>1?$
I know that there are many unanswered questions and conjectures when it comes to additive bases (of order $k=2$) of $\mathbb{N}.$ However, for the question I have in mind, I think it should be elementary to prove it, but I am having trouble doing…
Adam Rubinson
- 24,300
1
vote
0 answers
Large sets and Erdős-discrepancy
Large Sets
Erdos conjecture
I have a conjecture that is stronger than the Erdos discrepancy conjecture, can someone think of a counter example?
Let $S$ be any large set and let $(x_1,x_2,...)$ be any infinite $\pm 1$ sequence, then for any integer…
AndroidBeginner
- 702
1
vote
1 answer
Do large sets have this specific type of self-similarity?
Suppose $(a_n)_{n\in\mathbb{N}}$ is a strictly increasing sequence of
positive integers such that $\displaystyle\sum_{n\in\mathbb{N}}
\frac{1}{a_n}$ diverges, i.e. $(a_n)_{n\in\mathbb{N}}$ is "large".
Is it true that for every $k\in\mathbb{N},$…
Adam Rubinson
- 24,300
0
votes
2 answers
Has it been proven that, if $\ y_n = x_{n+1} - x_n\ $ is non-decreasing, then $\ x_n\ $ cannot be a counter-example to Erdős Conjecture?
I'm trying to find a subset $\ A\ $ of $\ \mathbb{N}\ $ that disproves Erdős Conjecture on Arithmetic progressions.
If we instead write $\ A\ $ as a (strictly) increasing sequence of integers, $\ (x_n)_n.\ $ My question is, has it been proven that,…
Adam Rubinson
- 24,300