Browse Problems

Is there an infinite sequence of distinct Gaussian primes $x_1,x_2,\ldots$ such that $\lvert x_{n+1}-x_n\rvert \ll 1$?

It is conjectured that $f(k) \ll (\log k)^O(1)$.

Erdos [Er55d] proved $f(k) < 3 \frac{k}{\log k}$ for sufficiently large $k$.

Jutila [Ju74], and Ramachandra--Shorey [RaSh73] proved a stronger upper bound $f(k) \ll \frac{\log \log \log k}{\log \log k} \frac{k}{\log k}$.

Sylvester and Schur [Er34] proved that every set of $k$ consecutive integers greater than $k$ contains an integer divisible by a prime greater than $k$, i.e. not $(k+1)$-smooth.

Erdős asks in [Er75b] if for every 2-coloring of ℝ, there is an uncountable set $A ⊆ ℝ$ such that all sums $a + b$ for $a, b ∈ A, a ≠ …

In fact, in both [Ko16] and [SWCol] a generalized example for $k$-sums is constructed.

Does the set `{n | u n < u (n+1)}` have positive natural density?

Erdős and Prachar proved that the set `{n | u n > u (n+1)}` has positive natural density (see [ErPr61]).

Erdős asked whether there are infinitely many solutions to `uₙ > uₙ₊₁ > uₙ₊₂`.

Erdős asked whether there are infinitely many solutions to `uₙ < uₙ₊₁ < uₙ₊₂`.

Erdős and Prachar proved `∑_{pₙ < x} |u (n+1) - u n| ≍ (log x)^2` (see [ErPr61]). We encode `∑_{pₙ < x}` as a sum over `n < Nat.primeCounting' x` (the number …

Does every convex polygon have a vertex with no other 4 vertices equidistant from it?

Let `p(a, d)` be the least prime congruent to `a (mod d)`. Does there exist a constant `c > 0` such that for all large `d`, `p(a, d) > (1 + c) …

Erdős [Er49c] proved that the statement in `erdos_971` holds for infinitely many values of `d`. [Er49c] Erdős, P., _On some applications of Brun's method_. Acta Univ. Szeged. Sect. Sci. Math. (1949), 57--63.

Erdős [Er49c] proved that for any `ε > 0` we have `p(a, d) < ε * φ(d) * log d` for `≫_ε φ(d)` many values of `a` (for all large `d`). [Er49c] …

**Erdős problem 972.** Let $\alpha > 1$ be irrational. Are there infinitely many primes $p$ such that $\lfloor p\alpha \rfloor$ is also prime?

For an irreducible polynomial $f \in \mathbb{Z}[x]$ with $f(n) \ge 1$ for sufficiently large $n$, does there exists a constant $c = c(f) > 0$ such that $\sum_{n \le x} \tau(f(n)) \approx …

More concrete example for $f(n) = n^2 + 1$, where the asymptote is $\sum_{n \le x} \tau(n^2 + 1) \sim \frac{3}{\pi} x \log x + O(x)$. See Tao's blog [T].

When $f$ is an irreducible quadratic polynomial, the question is answered first by Hooley [Ho63]. More compact expression of the constant in terms of Hurwitz class numbers (when $a = 1$) is …

The correctness of the growth rate is shown in [Va39] (lower bound) and [Er52b] (upper bound).

Does `n ^ 4 + 2` represent infinitely many squarefree numbers?

Let `f ∈ ℤ[X]` be an irreducible polynomial. Suppose that the degree `k` of `f` is larger than `2` and is not equal to a power of `2`. Then the set of …

Let `f ∈ ℤ[X]` be an irreducible polynomial. Suppose that the degree `k` of `f` is larger than `2`, and `f n` have no fixed `(k - 1)`-th power divisors other than …

If the degree `k` of `f` is larger than or equal to `9`, then the set of `n` such that `f n` is `(k - 2)`-th power free has infinitely many elements. …

Is it true that the set of `n` such that `f n` is `(k - 2)`-th power free has infinitely many elements?

Let $k ≥ 2$, and let $f_k(n)$ count the number of solutions to $n = p_1^k + \dots + p_k^k$, where the $p_i$ are prime numbers. Is it true that $\limsup f_k(n) …

Erdős [Er37b] proved that if $f_2(n)$ counts the number of solutions to $n = p_1^2 + p_2^2$, where $p_1$ and $p_2$ are prime numbers, then $\limsup f_2(n) = \infty$. [Er37b] Erdős, Paul, …

Erdős (unpublished)

Erdős also conjectured that there is a $k$ for which every convex polygon has a vertex with no other $k$ vertices equidistant from it.

Erdős originally conjectured this (in [Er46b]) with no 3 vertices equidistant, but Danzer found a convex polygon on 9 points such that every vertex has three vertices equidistant from it (but this …

Fishburn and Reeds [FiRe92] have found a convex polygon on 20 points such that every vertex has three vertices equidistant from it (and this distance is the same for all vertices). [FiRe92] …

Fishburn and Reeds [FiRe92] also proved that the smallest $n$ for which there exists a convex $n$-gon and a cut $\{A, B\}$ of its vertices such that $|\{b \in B : d(a, …

If $n$ distinct points in $\mathbb{R}^2$ form a convex polygon then some vertex has at least $\lfloor\frac{n}{2}\rfloor$ different distances to other vertices.

Is it true that, for every prime $p$, there is a prime $q \leq p$ which is a primitive root modulo $p$?

Heath-Brown proved that at least one of 2, 3, or 5 is a primitive root for infinitely many primes $p$.

For sufficiently large n, is it the case that any set of n points with minimum distance $1$ that minimizes diameter must contain an equilateral triangle of side length 1?

The following theorem is proved in [Ma66].

Does there exists a positive constant `C` such that for all `f ∈ L²[0,1]` and all lacunary sequences `n`, if `‖f - fₖ‖₂ = O(1 / log log log k ^ C)`, …

Is it true that, for every $\alpha$, the sequence $\{ \alpha p_n\}$ is not well-distributed, if $p_n$ is the sequence of primes?

He also claimed in [Er64b] to have proved that there exists an irrational $\alpha$ for which $\{\alpha p_n\}$ is not well-distributed. He later retracted this claim in [Er85e], saying "The theorem is …

Erdős proved that, if $n_k$ is a lacunary sequence, then the sequence $\{ \alpha n_k\}$ is not well-distributed for almost all $\alpha$.

The set is known to be infinite. In [Er77c] Erdős credits Schinzel with proving that there are infinitely many odd integers not of this form, but gives no reference. [Er77c] Erdős, P., …

The set is known to be infinite. In [Er77c] Erdős credits Schinzel with proving that there are infinitely many odd integers not of this form, but gives no reference. [Er77c] Erdős, P., …

Erdős and Szekeres conjectured that, apart from a finite exceptional set of triples `(n, i, j)`, one can always take `p > i` in the prime divisor statement.

Erdős and Turán [ErTu41] proved the upper bound of 1. [ErTu41] Erdős, P. and Turán, P., On a problem of Sidon in additive number theory, and on some related problems. J. London …

It is conjectured that the correct bound is $$ |E(r)| = O\left(r^{1/2 + o(1)}\right) $$ [Ha59] Hardy, G. H. (1959). _Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work_(3rd ed.). …