|
905df96a |
Can a sub-sum of reciprocals approach 1 from below within e^{-cK} once the mass exceeds K? (Erdős #312) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
d07f2072 |
Closest a distinct-unit-fraction sub-sum can get to 1: is δ(N) = e^{-(c+o(1))N}? (Erdős #311) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
49a51261 |
Represent every a/b (b squarefree) as a sum of distinct 1/(pq) with p,q distinct primes (Erdős #306) |
OPEN |
0 inv |
3.0 |
3.5 |
29d ago |
|
619bae4c |
Longest shortest Egyptian-fraction expansion: estimate N(b), is N(b) ≪ log log b? (Erdős #304) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
f5dd72db |
Largest subset of {1,…,N} with no 1/a = 1/b + 1/c: estimate f(N) (Erdős #302) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
a69b2f1f |
Largest subset of {1,…,N} with no 1/a equal to a sum of distinct 1/b_i: estimate f(N) (Erdős #301) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
4ac8f68c |
Do the first $N$ cubes contain a Sidon set of size $\gg N$? (Erdős #1206) |
OPEN |
0 inv |
3.0 |
3.5 |
29d ago |
|
c48e9d1c |
Largest Sidon subset guaranteed in every N-point real set: is $\ell(N)\sim N^{1/2}$? (Erdős #530) |
ACTIVE |
1 inv |
3.0 |
2.0 |
18d ago |
|
00af2f59 |
Largest subset of {1,...,N} with all pairwise products distinct: pin the constant in $F(N)$ (Erdős #425) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
e9c8aed9 |
Does $k(N)-(e-1)N\to\infty$? Terms needed for a unit-fraction sum to $1$ with denominators $\geq N$ (Erdős #295) |
OPEN |
0 inv |
2.5 |
3.0 |
29d ago |
|
74ee34bf |
Growth of $v(k)$, the least integer missing from every $k$-term unit-fraction representation of $1$ (Erdős #293) |
OPEN |
0 inv |
2.5 |
2.0 |
29d ago |
|
dbce7ae6 |
For all large $k$, can $1$ be written as a sum of reciprocals over $k$ disjoint integer intervals? (Erdős #289) |
OPEN |
0 inv |
2.0 |
3.0 |
29d ago |
|
bf0af53b |
Are there only finitely many pairs of integer intervals whose reciprocal sums total an integer? (Erdős #288) |
OPEN |
0 inv |
2.0 |
3.5 |
29d ago |
|
ead15314 |
Does the odd-greedy Egyptian-fraction algorithm always terminate for odd-denominator rationals? (Erdős #282) |
OPEN |
0 inv |
3.0 |
3.5 |
29d ago |
|
9077a647 |
Is there an infinite composite-coordinate path in the visible-lattice-point graph? (Erdős #1212) |
OPEN |
0 inv |
2.0 |
3.0 |
29d ago |
|
6c54dfc0 |
Do most integers n have a large prime factor within a bounded window n,...,n+k? (Erdős #1201) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
26eff08f |
Can primes of bounded reciprocal sum cover every integer below x by congruences? (Erdős #1200) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
9bd810f8 |
Is the completely-multiplicative random partial sum a.s. unbounded relative to N^{1/2}? (Erdős #1144) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
a8c2db46 |
Which sequences b_n admit a primitive sequence a_n growing no faster than b_n? (Erdős #892) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
7d0410c7 |
How long can the primitive-set saturation game be forced to last? (Erdős #872) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
7ef01369 |
Estimate the Folkman numbers F(k): a monochromatic k-set with all subset sums one colour (Erdős #531) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
a21d6917 |
Does the Rademacher random multiplicative partial sum obey an iterated-logarithm law? (Erdős #520) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
3dcfcd6f |
Is $f(n,k)=(1-\rho(\alpha)+o(1))k$ for the count of $n+i$ with prime factor $>k$? (Erdős #1184) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
66bd02c7 |
Estimate $F_k(p_1,\ldots,p_u)$: multiples of some $p_i$ forced in every length-$k$ interval (Erdős #1143) |
OPEN |
0 inv |
2.0 |
2.5 |
29d ago |
|
262a2c1a |
Integers $n>105$ with $n-2^k$ prime for all $1<2^k<n$: any, or infinitely many? (Erdős #1142) |
OPEN |
0 inv |
2.5 |
2.0 |
29d ago |
|
adc11e2a |
Gaps between integers with at most two prime factors: is $\limsup (u_{k+1}-u_k)/\log k=\infty$? (Erdős #1139) |
OPEN |
0 inv |
2.0 |
3.0 |
29d ago |
|
07a1e5a7 |
Infinitely many primes $p$ with every $p-k!$ composite (for $k!<p$)? (Erdős #1059) |
OPEN |
0 inv |
2.0 |
3.0 |
29d ago |
|
f7f07f6f |
Erdős–Selfridge prime classes: infinitely many primes per class, and growth of $p_r^{1/r}$ (Erdős #1055) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
aca7fd16 |
Growth of $f(n)=\sum_{p<n}1/(n-p)$: liminf, limsup, and an $o(\log\log n)$ bound (Erdős #950) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
4ca68a54 |
Restricted prime-factor counts over consecutive integers: a liminf bound and a limsup law (Erdős #890) |
OPEN |
0 inv |
2.5 |
2.0 |
29d ago |
|
3f2bb9fd |
Smallest even value missing from the first $x$ prime gaps: does $r(x)\to\infty$? (Erdős #853) |
OPEN |
0 inv |
2.5 |
3.0 |
29d ago |
|
cb9bf76e |
Longest run of distinct consecutive prime gaps: estimate $h(x)$ (Erdős #852) |
OPEN |
0 inv |
3.0 |
3.5 |
29d ago |
|
378c569f |
Is there a composite $n+k$ with least prime factor $p(n+k)>k^2$ for all large $n$? (Erdős #681) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
346a8881 |
Least prime factor spikes: is $p(n+k)>k^2+1$ solvable for every large $n$? (Erdős #680) |
OPEN |
0 inv |
3.0 |
2.5 |
29d ago |
|
54a1b295 |
Is there $f(n)\to\infty$ with a composite $m$ satisfying $n+f(n)<m<n+p(m)$? (Erdős #463) |
OPEN |
0 inv |
2.0 |
2.5 |
29d ago |
|
3b24ada0 |
Is the least-prime-factor sum $\sum p(n)/n$ over every short window $\gg 1$? (Erdős #462) |
OPEN |
0 inv |
2.0 |
3.0 |
29d ago |
|
71b06748 |
Distinct $t$-smooth components in a window of length $t$: is $f(n,t)\gg t$? (Erdős #461) |
OPEN |
0 inv |
2.5 |
3.0 |
29d ago |
|
67afa874 |
A positive relative-density set $A$ with all $n-a$ prime for infinitely many $n$ (Erdős #428) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
9bf3a6ac |
Does $\{p+\lfloor C^k\rfloor\}$ have positive density for every $C>1$? (Erdős #244) |
OPEN |
0 inv |
2.5 |
1.5 |
29d ago |
|
396ead69 |
Runs of $>c_1\log x$ consecutive primes with all gaps $>c_2$: must they always exist? (Erdős #238) |
OPEN |
0 inv |
2.5 |
1.5 |
29d ago |
|
7a1c8d11 |
Is the number of representations $n=p+2^k$ always $o(\log n)$? (Erdős #236) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |
|
03868985 |
Do normalized prime gaps have a continuous limiting distribution function? (Erdős #234) |
OPEN |
0 inv |
3.0 |
1.5 |
29d ago |
|
5fef66f5 |
Prove $\sum_{n\le N}(p_{n+1}-p_n)^2\ll N(\log N)^2$ for prime gaps (Erdős #233) |
OPEN |
0 inv |
3.5 |
1.5 |
29d ago |
|
2b504461 |
Are there infinitely many cluster primes? (Erdős #17) |
ACTIVE |
1 inv |
3.0 |
3.5 |
23d ago |
|
04882410 |
Does the alternating prime series $\sum(-1)^n n/p_n$ converge? (Erdős #15) |
OPEN |
0 inv |
2.5 |
1.5 |
29d ago |
|
fa409647 |
Limit points of normalized prime gaps: is $S=[0,\infty]$ for $(p_{n+1}-p_n)/\log n$? (Erdős #5) |
OPEN |
0 inv |
3.5 |
1.5 |
29d ago |
|
26339f6f |
Coprime sets in $[1,n)$: is $\sum_{a\in A}1/(n-a)\leq\sum_{p<n}1/p+O(1)$? (Erdős #1210) |
OPEN |
0 inv |
2.5 |
2.5 |
29d ago |
|
971b359f |
Diameter of admissible k-tuples: is $A(k)\sim k\log k$? (and estimate the mean $B(k)$) (Erdős #1204) |
OPEN |
0 inv |
3.5 |
2.0 |
29d ago |
|
37310009 |
Is every large integer a sum of at most $r+1$ many $r$-powerful numbers? (Erdős #1107) |
OPEN |
0 inv |
3.0 |
3.0 |
29d ago |
|
49656b48 |
Density of sums of three $k$-th powers: is $f_{k,3}(x)\gg x^{3/k}$? (Erdős #325) |
OPEN |
0 inv |
3.0 |
2.0 |
29d ago |