6 problems

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Erdős Problem #124 — For any $d\geq 1$ and $k\geq 0$ let $P(d,k)$ be the set of integers which are the sum of distinct powers $d^i$ with $i\geq k$ .
About 30 years old
For any $d\geq 1$ and $k\geq 0$ let $P(d,k)$ be the set of integers which are the sum of distinct powers $d^i$ with $i\geq k$ . Let $3\leq d_1<d_2<\cdots <d_r$ be integers such that…
claimed progressErdősbase-representations number-theory
019
Erdős Problem #125 — Let $A = \{ \sum\epsilon_k3^k : \epsilon_k\in \{0,1\}\}$ be the set of integers which have only the digits $0,1$ when written base $3$ , and $B=\{ \sum\epsilon_k4^k : \epsilon_k\in \{0,1\}\}$ be the se…
At least 30 years old
Let $A = \{ \sum\epsilon_k3^k : \epsilon_k\in \{0,1\}\}$ be the set of integers which have only the digits $0,1$ when written base $3$ , and…
solvedFormal statementErdősbase-representations number-theory
016
Erdős Problem #730 — Are there infinitely many pairs of integers $n\neq m$ such that $\binom{2n}{n}$ and $\binom{2m}{m}$ have the same set of prime divisors?
About 51 years old
Are there infinitely many pairs of integers $n\neq m$ such that $\binom{2n}{n}$ and $\binom{2m}{m}$ have the same set of prime divisors?
openFormal statementErdősbase-representations binomial-coefficients number-theory
07
Erdős Problem #406 — Is it true that there are only finitely many powers of $2$ which have only the digits $0$ and $1$ when written in base $3$ ?
About 47 years old
Is it true that there are only finitely many powers of $2$ which have only the digits $0$ and $1$ when written in base $3$ ?
openFormal statementErdősbase-representations number-theory
010
Erdős Problem #376 — Are there infinitely many $n$ such that $\binom{2n}{n}$ is coprime to $105$ ?
At least 45 years old
Are there infinitely many $n$ such that $\binom{2n}{n}$ is coprime to $105$ ?
openFormal statementErdősbase-representations binomial-coefficients number-theory
07
The lexicographic characterization of the sequences $S_k$
About 7 years old
Let $\mathcal{S}_k$ be the sequence written in base $3/2$ , and let $S_k$ denote the result of interpreting that representation in base $3$ . The lexicographic characterization conje…
openarithmetic-progressions base-representations
014

6 problems

Erdős Problem #124 — For any d≥1d\geq 1d≥1 and k≥0k\geq 0k≥0 let P(d,k)P(d,k)P(d,k) be the set of integers which are the sum of distinct powers did^idi with i≥ki\geq ki≥k.

Erdős Problem #730 — Are there infinitely many pairs of integers n≠mn\neq mn=m such that (2nn)\binom{2n}{n}(n2n​) and (2mm)\binom{2m}{m}(m2m​) have the same set of prime divisors?

Erdős Problem #406 — Is it true that there are only finitely many powers of 222 which have only the digits 000 and 111 when written in base 333?

Erdős Problem #376 — Are there infinitely many nnn such that (2nn)\binom{2n}{n}(n2n​) is coprime to 105105105?

The lexicographic characterization of the sequences SkS_kSk​

Erdős Problem #124 — For any $d\geq 1$ and $k\geq 0$ let $P(d,k)$ be the set of integers which are the sum of distinct powers $d^i$ with $i\geq k$ .

Erdős Problem #730 — Are there infinitely many pairs of integers $n\neq m$ such that $\binom{2n}{n}$ and $\binom{2m}{m}$ have the same set of prime divisors?

Erdős Problem #406 — Is it true that there are only finitely many powers of $2$ which have only the digits $0$ and $1$ when written in base $3$ ?

Erdős Problem #376 — Are there infinitely many $n$ such that $\binom{2n}{n}$ is coprime to $105$ ?

The lexicographic characterization of the sequences $S_k$