Questions tagged [erdos-conjecture]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
3 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 $$ 2^x=12(5^{y_1}+ \dots +5^{y_k}) + 5^z - 1 $$ ...
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)...
0 votes
0 answers

Find largest $k$ given $n$: one number is chosen from each interval $[nj,n(j+1)),\ j\in\{1,\ldots,k\};$ no three of these numbers form a $3$-term A.P.

Given $\ n,k\in\mathbb{N},\ $ one integer is chosen from each interval $\ [\ n(j-1),\ nj\ ), \ j\in\{1,\ldots,k\},\ $ such that no three of these numbers form a three-term arithmetic progression. What ...
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(...