Browse Problems

In 1950, Erdős [Er50c] proved the lower bound $$\log \log b \ll N(b)$$. [Er50c] Erdős, P., Az ${1}/{x_1} + {1}/{x_2} + \ldots + {1}/{x_n} =A/B$ egyenlet eg\'{E}sz sz\'{A}m\'{u} megold\'{A}sairól. Mat. Lapok (1950), …

In 1950, Erdős [Er50c] proved the upper bound $$N(b) \ll \log b / \log \log b$$. [Er50c] Erdős, P., Az ${1}/{x_1} + {1}/{x_2} + \ldots + {1}/{x_n} =A/B$ egyenlet eg\'{E}sz sz\'{A}m\'{u} megold\'{A}sairól. …

In 1985 Vose [Vo85] proved the upper bound $$N(b) \ll \sqrt{\log b}$$. [Vo85] Vose, Michael D., Egyptian fractions. Bull. London Math. Soc. (1985), 21-24.

Let $\frac a b\in \mathbb{Q}_{>0}$ with $b$ squarefree. Are there integers $1 < n_1 < \dots < n_k$, each the product of two distinct primes, such that $\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$?

Every positive integer can be expressed as an Egyptian fraction where each denominator is the product of three distinct primes.

Are there two finite set of primes $P$ and $Q$ such that $$ 1 = \left( \sum_{p \in P} \frac{1}{p} \right) \left( \sum_{q \in Q} \frac{1}{q} \right) $$ ? Asked by Barbeau …

There are no examples known of the weakened coprime version if we insist that $1\not\in P\cup Q$.

Does there exist a constant `c > 0` such that, for any `K > 1`, whenever `A` is a sufficiently large finite multiset of integers with $\sum_{n \in A} 1/n > K$ …

Are there infinitely many pairs `(m, P)` where `m ≥ 2` is an integer and `P` is a set of distinct primes such that the following equation holds: $\sum_{p \in P} \frac{1}{p} …

It is conjectured that the set of primary pseudoperfect numbers is infinite.

Is it true that if $A \subseteq \mathbb{N}\setminus\{1\}$ is a finite set with $\sum_{n \in A} \frac{1}{n} < 2$ then there is a partition $A=A_1 \sqcup A_2$ such that $\sum_{n \in A_i} …

More generally, Sándor shows that for any $n \ge 2$ there exists a finite set $A \subseteq \mathbb{N}\setminus\{1\}$ with $\sum_{k \in A} \frac{1}{k} < n$, and no partition into $n$ parts each …

Is there some constant $c>0$ such that for every $n\geq 1$ there exists some $\delta_k\in \{-1,0,1\}$ for $1\leq k\leq n$ with \[0< \left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert < \frac{c}{2^n}?\]

Is it true that for sufficiently large $n$, for any $\delta_k\in \{-1,0,1\}$, \[\left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert > \frac{1}{[1,\ldots,n]}\] whenever the left-hand side is not zero?

For any set `A` containing exactly one even number, `A` does not have property `P₁`. Sattler [Sa82] credits this observation to Erdős, who presumably found this after [ErGr80].

Every infinite arithmetic progression has property `P₁`. This is proved in [Sa82b].

Sattler proved in [Sa75] that the set of odd numbers has property `P₁`.

There exists a set `A` with positive density that does not have property `P₁`. #TODO: prove this lemma by assuming `erdos_318.contain_single_even`.

Does the set of squares excluding 1 have property `P₁`?

`ℕ` has property `P₁`. This is proved in [ErSt75].

What is the size of the largest $A\subseteq\{1, \dots, N\}$ such that there is a function $\delta : A \to \{-1, 1\}$ such that $$ \sum_{n\in A} \frac{\delta n}{n} = 0 $$ …

Let $c(N)$ be the size of the largest $A\subseteq\{1, \dots, N\}$ such that there is a function $\delta : A \to \{-1, 1\}$ such that $$ \sum_{n\in A} \frac{\delta n}{n} = 0 …

Let $c(N)$ be the size of the largest $A\subseteq\{1, \dots, N\}$ such that there is a function $\delta : A \to \{-1, 1\}$ such that $$ \sum_{n\in A} \frac{\delta n}{n} = 0 …

Let $c(N)$ be the size of the largest $A\subseteq\{1, \dots, N\}$ such that there is a function $\delta : A \to \{-1, 1\}$ such that $$ \sum_{n\in A} \frac{\delta n}{n} = 0 …

Adenwalla has observed that a lower bound (on the maximum size of $A$) of $$ |A| \geq (1 - \frac{1}{e} + o(1))N $$ follows from the main result of Croot [Cr01]. [Cr01] …

Does there exist a set $A \subseteq \mathbb{N}$ such that $|A \cap \{1, \ldots, N\}| = o((\log N)^2)$ and every sufficiently large integer can be written as $p + a$ for some …

Let $R(N)$ be the size of the largest $A\subseteq\{1, ..., N\}$ such that all sums $\sum_{n\in S} \frac{1}{n}$ are distinct for $S\subseteq A$. What is $R(N)$?

Let $R(N)$ be the size of the largest $A\subseteq\{1, ..., N\}$ such that all sums $\sum_{n\in S} \frac{1}{n}$ are distinct for $S\subseteq A$. Find the simplest $g(N)$ such that $R(N) = O(g(N))$.

Let $R(N)$ be the size of the largest $A\subseteq\{1, ..., N\}$ such that all sums $\sum_{n\in S} \frac{1}{n}$ are distinct for $S\subseteq A$. Find the simplest $g(N)$ such that $R(N) = o(g(N))$.

Let $R(N)$ be the size of the largest $A\subseteq\{1, ..., N\}$ such that all sums $\sum_{n\in S} \frac{1}{n}$ are distinct for $S\subseteq A$. What is $\Theta(R(N))$?

Let $R(N)$ be the maximal such size. Results of Bleicher and Erdős from [BlEr75] and [BlEr76b] imply that $$ \frac{N}{\log N} \prod_{i=3}^{k} \log_i N \le R(N), $$ valid for any $k \ge …

Let $R(N)$ be the maximal such size. Results of Bleicher and Erdős from [BlEr75] and [BlEr76b] imply that $$ R(N) \le \frac{1}{\log 2} \log_r N \left( \frac{N}{\log N} \prod_{i=3}^{r} \log_i N \right), …

Does there exist a polynomial $f(x)\in\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with $a < b$ nonnegative integers are distinct?

Probably $f(x) = x^5$ has the property that the sums $f(a)+f(b)$ with $a < b$ nonnegative integers are distinct.

Writing $f_{k, 3}(x)$ for the number of integers $\leq x$ which are the sum of three $k$th powers, is it true that $f_{k, 3}(x) \gg x ^ (3 / k)$?

Writing $f_{k, 3}(x)$ for the number of integers $\leq x$ which are the sum of three $k$th powers, is it even true that $f_{k, 3}(x) \gg_{\epsilon} x ^ (3 / k - …

For $k = 3$, the best known is due to Wooley [Wo15] [Wo15] Wooley, Trevor D., Sums of three cubes, II. Acta Arith. (2015), 73-100.

Let $A \subset \mathbb{N}$ be an additive basis of order 2. Must there exist $B = \{b_1 < b_2 < \dots\} \subseteq A$ which is also a basis such that $\lim_{k\to\infty} \frac{b_k}{k^2}$ …

**Erdős Problem 329.** Let `A ⊆ ℕ` be a Sidon set. How large can `lim sup_{N → ∞} |A ∩ {1,…,N}| / N^{1/2}` be?

The converse: if the maximum density is 1, then any finite Sidon set can be embedded in a perfect difference set.

Erdős proved that upper density `1 / 2` can be attained; in particular, there exists a Sidon set whose upper density is *at least* `1 / 2`.

If any finite Sidon set can be embedded in a perfect difference set, then the maximum density would be 1.

Erdős proved in [Erd54] that there exists an additive complement $A$ to the primes with $|A \cap \{1, \ldots, N\}| = O((\log N)^2)$.

Must every additive complement $A$ to the primes satisfy $\liminf_{N \to \infty} \frac{|A \cap \{1, \ldots, N\}|}{\log N} > 1$?

Can the bound $O(\log N)$ be achieved for an additive complement to the primes? [Guy04] writes that Erdős offered \$50 for the solution.

Ruzsa proved that any additive complement $A$ to the primes must satisfy $\liminf_{N \to \infty} \frac{|A \cap \{1, \ldots, N\}|}{\log N} \geq e^\gamma$, where $\gamma$ is the Euler-Mascheroni constant.

Let `A ⊆ ℕ` be a set such that every integer can be written as `n^2 + a` for some `a` in `A` and `n ≥ 0`. What is the smallest possible …

Does there exist a minimal basis $A \subset \mathbb{N}$ with positive density such that, for any $n \in A$, the (upper) density of integers which cannot be represented without using $n$ is …

Let $A,B\subseteq \mathbb{N}$ such that for all large $N$\[\lvert A\cap \{1,\ldots,N\}\rvert \gg N^{1/2}\]and\[\lvert B\cap \{1,\ldots,N\}\rvert \gg N^{1/2}.\] Is it true that there are infinitely many solutions to $a_1-a_2=b_1-b_2\neq 0$ with $a_1,a_2\in A$ …