|
bf9e3bb8 |
A polynomial whose pairwise sums are all distinct (a polynomial Sidon set): does one exist? (Erdős #324) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
120a987f |
Density of sums of $k$-th powers: is $f_{k,k}(x)\gg x^{1-\epsilon}$ and $f_{k,m}(x)\gg x^{m/k}$? (Erdős #323) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
93c587af |
Representations as sums of $k$ many $k$-th powers: can the count exceed $n^c$ infinitely often? (Erdős #322) |
OPEN |
0 inv |
3.5 |
2.0 |
29d ago |
|
4b71a256 |
Prove the weighted shift-maximum $F(n)=\max_k\omega(n+k)\log\log k/\log k$ diverges (Erdős #1203) |
OPEN |
0 inv |
2.5 |
2.0 |
29d ago |
|
5603169c |
Least prime missing from a run of $\log n$ consecutive integers: below $(1-c)(\log n)^2$? (Erdős #1181) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
c7de7120 |
Are the $3$-smooth numbers $\{2^m3^n\}$ an essential component? (Erdős #1146) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
212bf571 |
Additive functions that rarely decrease at $n\mapsto n+1$: must they be $c\log n$? (Erdős #1122) |
OPEN |
0 inv |
3.0 |
1.0 |
29d ago |
|
f68cbd7e |
Largest subset of $\{1,\ldots,N\}$ whose pairwise sums are all squarefree (Erdős #1109) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
696cae75 |
Number of distinct primes dividing the product of the first $n$ partition numbers (Erdős #1106) |
OPEN |
0 inv |
3.0 |
3.5 |
29d ago |
|
914bd9a4 |
Growth rate of an infinite sequence whose pairwise sums are all squarefree (Erdős #1103) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
5e962925 |
Densities of EHS numbers and Pillai primes: do the counting ratios converge, and to what? (Erdős #1074) |
OPEN |
0 inv |
2.5 |
3.5 |
29d ago |
|
46dadb9f |
Count composite $u$ with $n!+1\equiv0\pmod u$ for some $n$: is $A(x)\leq x^{o(1)}$? (Erdős #1073) |
OPEN |
0 inv |
2.5 |
2.5 |
29d ago |
|
e7091b5b |
Least $n$ with $n!+1\equiv0\pmod p$: is $f(p)=p-1$ infinitely often, and $f(p)=o(p)$ a.e.? (Erdős #1072) |
OPEN |
0 inv |
2.5 |
3.5 |
29d ago |
|
f14bcb58 |
Are there infinitely many primes $p=2^kq+1$ (or $2^k3^\ell q+1$) with $q$ prime? (Erdős #1065) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
ce4d30fc |
Estimate $n_k$, least $n\geq 2k$ with $n-i\mid\binom{n}{k}$ for all but one $i<k$ (Erdős #1063) |
OPEN |
0 inv |
2.5 |
3.5 |
29d ago |
|
dc5ca039 |
Largest $A\subseteq[n]$ with no element dividing two others: is $\lim f(n)/n$ irrational? (Erdős #1062) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
e725baa9 |
Bound the multiplicity of $k\sigma(k)=n$: is the number of solutions $n^{o(1/\log\log n)}$? (Erdős #1060) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
6b17bfd5 |
Carmichael numbers: is the count $C(x)=x^{1-o(1)}$? (Erdős #1057) |
OPEN |
0 inv |
3.5 |
2.0 |
29d ago |
|
47340079 |
Consecutive integer blocks each with product $\equiv 1 \pmod p$: possible for every $k$? (Erdős #1056) |
OPEN |
0 inv |
2.0 |
4.0 |
29d ago |
|
d7330f1b |
Multiply perfect numbers: must the multiplier satisfy $k=o(\log\log n)$? (Erdős #1053) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
e6a5cff0 |
Are there only finitely many unitary perfect numbers? (Erdős #1052) |
OPEN |
0 inv |
2.0 |
2.0 |
29d ago |
|
277a09f2 |
Order of the longest similarly-ordered run of Farey fractions: is $f(n)\sim cn$? (Erdős #1005) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
902407ee |
A run of $(\log x)^c$ consecutive integers with all distinct totient values? (Erdős #1004) |
OPEN |
0 inv |
2.0 |
2.5 |
29d ago |
|
ed4d7f45 |
Are there infinitely many $n$ with $\phi(n)=\phi(n+1)$? (Erdős #1003) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
e8343875 |
A prime primitive root below every prime: does one always exist? (Erdős #985) |
OPEN |
0 inv |
2.5 |
3.5 |
29d ago |
|
80a77976 |
Estimate $f(k,n)$: primes needed to over-cover a $k$-subset of $\{1,\ldots,n\}$ (Erdős #983) |
OPEN |
0 inv |
2.0 |
2.0 |
29d ago |
|
633a2336 |
Unbounded representation counts as sums of $k$ prime $k$-th powers: is $\limsup f_k(n)=\infty$? (Erdős #979) |
OPEN |
0 inv |
3.0 |
3.5 |
29d ago |
|
181ca648 |
Are there infinitely many $n$ with $n^4+2$ squarefree? Power-free values of polynomials (Erdős #978) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
e0dd0d29 |
Greatest prime factor of $\prod_{m\le n}f(m)$: is it $\gg n^{1+c}$ for irreducible $f$? (Erdős #976) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
0b273a83 |
Infinitely many primes $p$ with $\lfloor p\alpha\rfloor$ also prime, for irrational $\alpha>1$? (Erdős #972) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
0587beee |
Are there $\gg\phi(d)$ residues $a$ with least prime $p(a,d)>(1+c)\phi(d)\log d$? (Erdős #971) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
e034b1d4 |
Order of magnitude of Jacobsthal's function $h(k)$: is $h(k)\ll k^2$? (Erdős #970) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
16bd50a7 |
Order of magnitude of the error term $E(x)$ in the count of squarefree integers (Erdős #969) |
OPEN |
0 inv |
3.5 |
2.0 |
29d ago |
|
a5f9fd41 |
Does {n : p_n/n < p_{n+1}/(n+1)} have positive density? (Erdős #968) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
96ee4052 |
Largest guaranteed dissociated subset f(n): is f(n) ≥ ⌊log₂ n⌋? (Erdős #963) |
ACTIVE |
2 inv |
3.0 |
2.0 |
15d ago |
|
5a9a3c15 |
Growth of k(n): runs of integers with a large prime factor > k (Erdős #962) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
abea354d |
Erdős–Granville–Pomerance–Spiro: does density 0 pull back to density 0 under s(n)? (Erdős #955) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
11aa123d |
Error term for Rosen's greedy B_2-type sequence: is R(x)=x+O(x^{1/4+o(1)})? (Erdős #954) |
OPEN |
0 inv |
2.0 |
3.5 |
29d ago |
|
35f2b18b |
Gaussian moat: is there an infinite bounded-step walk on Gaussian primes? (Erdős #952) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
c479ce46 |
Do Beurling generalised primes satisfy #{a_i ≤ x} ≤ π(x)? (Erdős #951) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
1168e89a |
Is the two-powerful-number representation function n^{o(1)}? (Erdős #943) |
OPEN |
0 inv |
2.0 |
2.5 |
29d ago |
|
a9ed455c |
Estimate $S(k)$, the least $x$ forcing dense $k$-runs each divisible by a prime $\leq x$ (Erdős #929) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
85b24440 |
Does the density of $n$ with $P(n)<n^\alpha$ and $P(n+1)<(n+1)^\beta$ exist? (Erdős #928) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
928bd37d |
Infinitely many $n$ with all exponents in the factorisation of $n(n+1)$ distinct? (Erdős #913) |
OPEN |
0 inv |
2.5 |
2.5 |
29d ago |
|
d3feaa35 |
Must every length-$p_1\cdots p_k$ interval contain an integer with $>k$ prime factors? (Erdős #891) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
136f5ccb |
Erdős–Selfridge: does the peak count of large 'new' prime factors $v_0(n)$ tend to infinity? (Erdős #889) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
9174225d |
Maximal sum of a pairwise-coprime subset of $\{1,\ldots,n\}$: is $G(n)>H(n)-n^{1+o(1)}$? (Erdős #879) |
OPEN |
0 inv |
3.0 |
3.5 |
29d ago |
|
ac9c766e |
Extremal order and coincidence of the prime-power functions $f(n)$ and $F(n)$ (Erdős #878) |
OPEN |
0 inv |
2.5 |
3.0 |
29d ago |
|
d81452b3 |
Sliding-window LCM counts of a sequence: can $F(A,X,k)<X^\epsilon$ be forced for some $k$? (Erdős #873) |
OPEN |
0 inv |
2.5 |
2.0 |
29d ago |
|
7ce72216 |
Growth of $H(n)$, least $l$ with $\gcd(k^n\!-\!1,l^n\!-\!1)=1$ for some $k<l$: is $H(n)=3$ i.o.? (Erdős #820) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |