Catalog
Every one of these is still open.
Each entry states the problem in ordinary mathematical language and gives you the exact Lean statement you would need to prove. Nothing is paraphrased, so what you read is what gets checked.
146 problems
- Is the upper density of the set of odd numbers that cannot be expressed as a prime plus two powers of 2 positive?
Combinatorics
Erdős 9
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- Bogdan Grechuk has observed that 1117175146 is not the sum of a prime and at most 3 powers of 2, and pointed out that parity considerations, coupled with the fact that there are many integers not the sum of a prime and 2 powers of 2 suggest that there exist infinitely many even integers which are not the sum…
Combinatorics
Erdős 10 - grechuk
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- Is every odd n>1 the sum of a squarefree number and a power of 2?
Number theory
Erdős 11
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- Is it true that ∑n=1∞(−1)npnn converges, where pn is the sequence of primes? Note: In the problem statement, pn is the n-th prime, indexed such that p1=2,p2=3,…. We 0-index here to reflect how Nat.nth works.
Number theory
Erdős 15
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- **Erdős Problem 17.** Are there infinitely many cluster primes?
Number theory
Erdős 17
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- If A⊆N is such that A+A contains all but finitely many integers then limsup1A∗1A(n)=∞.
Number theory
Erdős 28
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- Let
Number theory
Erdős 41
A ⊆ ℕbe an infinite set such that the triple sumsa + b + care all distinct fora, b, cinA(aside from the trivial coincidences). Is it true thatliminf n → ∞ |A ∩ {1, …, N}| / N^(1/3) = 0?- Attempts
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- Is ∑n=2∞n!−11 irrational?
Number theory
Erdős 68
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- F(n)/logn→∞asn→∞
Combinatorics
Erdős 82
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- Erdős [Er46] asked whether every set of n distinct points in R2 determines ≫lognn many distinct distances.
Convex and discrete geometry
Erdős 89
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- If n points in R2 form a convex polygon then there are O(n) many pairs which are distance 1 apart.
Convex and discrete geometry
Erdős 96
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- Is the diameter of A at least Cn for some constant C>0?
Convex and discrete geometry
Erdős 100
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- Let A⊆R be an infinite set. Must there be a set E⊆R of positive measure which does not contain any set of the shape a∗A+b for some a,b∈R and a=0?
Combinatorics
Erdős 120
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- Erdős says that f(n)=o(lognn) has never been proved.
Number theory
Erdős 126 - isLittleO
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- Erdős [Er82c] conjectures that, if k is fixed, then for all n sufficiently large and all positive integers m, there must be at least k distinct primes p such that p∣m(m+1)⋯(m+n) and yet p2 does not divide the right hand side. [Er82c] Erdős, Paul, "Miscellaneous problems in number theory".…
Number theory
Erdős 137 - multiple powerful factors
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- Are there 11 consecutive primes in arithmetic progression?
Combinatorics
Erdős 141 - eleven
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- Show that rk(N)=ok(N/logN), where rk(N) the largest possible size of a subset of {1,…,N} that does not contain any non-trivial k-term arithmetic progression.
Number theory
Erdős 142 - lower
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- Or ∑x∈Axlogx1<∞,
Number theory
Erdős 143 - part ii
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- Let A be a finite Sidon set and A+A={s1<⋯<st}. Is it true that t1∑1≤i<t(si+1−si)2→∞ as ∣A∣→∞?
Combinatorics
Erdős 153
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- Any graph on n vertices can be decomposed into O(n) many edge-disjoint cycles and edges.
Combinatorics
Erdős 184
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- Must every permutation of N, contain a monotone 4-term arithmetic progression?
Combinatorics
Erdős 196
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- In [Er79] Erdős says perhaps sn+1−sn≪logsn, but he is 'very doubtful'. [Er79] Erdős, Paul, __Some unconventional problems in number theory__. Math. Mag. (1979), 67-70.
Number theory
Erdős 208 - log bound
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- Is there a dense subset of ℝ^2 such that all pairwise distances are rational?
Convex and discrete geometry
Erdős 212
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- Let n≥4. Are there n points in R2, no three on a line and no four on a circle, such that all pairwise distances are integers?
Convex and discrete geometry
Erdős 213
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- The set of indices n for which a prime gap is preceeded by a larger or equal prime gap has a natural density of 21.
Number theory
Erdős 218 - ge
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- There are infintely many indices n such that the prime gap at n is equal to the prime gap at n+1. This is equivalent to the existence of infinitely many arithmetic progressions of length 3, see
Number theory
Erdős 218 - infinite equal prime gap
erdos_141.variants.infinite_three.- Attempts
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- The set of indices n for which a prime gap is followed by a larger or equal prime gap has a natural density of 21.
Number theory
Erdős 218 - le
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- A conjecture by Heath-Brown: The sum of squares of the first N gaps between consecutive primes behaves like N∗(logN)2.
Number theory
Erdős 233
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- Let f(n) count the number of solutions to n=p+2k for prime p and k≥0. Show that f(n)=o(logn).
Combinatorics
Erdős 236
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- More generally, Bose and Chowla [BoCh62] conjectured that the maximum size of A⊆{1,…,N} with all r-fold sums distinct (aside from the trivial coincidences) then ∣A∣∼N1/r.
Combinatorics
Erdős 241 - generalization
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- For every n>2 there exist distinct integers 1≤x<y<z such that n4=x1+y1+z1.
Number theory
Erdős 242
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- Schinzel conjectured (see [Si56]) the generalisation that, for any fixed a, if n is sufficiently large in terms of a then there exist distinct integers 1≤x<y<z such that na=x1+y1+z1.
Number theory
Erdős 242 - schinzel generalization
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- Let a1<a2<… be a sequence of integers such that limn→∞an−12an=1 and ∑an1∈Q. Then, for all sufficiently large n≥1, an=an−12−an−1+1.
Sequences and series
Erdős 243
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- Is ∑n2nϕ(n) irrational? Here ϕ is the Euler totient function.
Number theory
Erdős 249
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- Is ∑n=1∞2npn irrational? Here pn is the n-th prime (p1=2,p2=3,…).
Number theory
Erdős 251
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- Erdős Problem 252: irrationality of the sum for a given k.
Number theory
Erdős 252
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- Is n! an example of an irrationality sequence?
Number theory
Erdős 264 - part ii
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- Szabo asks whether the maximal t is given by 2N2+O(N)
Combinatorics
Erdős 272 - szabo strong
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- Let G be a group, and let A={a1G1,…,akGk} be a finite system of left cosets of subgroups G1,…,Gk of G. Herzog and Schönheim conjectured that if A forms a partition of G with k>1, then the indices [G:G1],…,[G:Gk] cannot be distinct.
Group theory
Erdős 274 - herzog schonheim
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- Let A⊆N be an infinite set and consider the following greedy algorithm for a rational x∈(0,1): choose the minimal n∈A such that n≥1/x and repeat with x replaced by x−n1. If this terminates after finitely many steps then this produces a representation of x as the sum…
Combinatorics
Erdős 282
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- It is conjectured that the set of primary pseudoperfect numbers is infinite.
Number theory
Erdős 313 - primary pseudoperfect are infinite
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- Probably f(x)=x5 has the property that the sums f(a)+f(b) with a<b nonnegative integers are distinct.
Number theory
Erdős 324 - quintic
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- Let A={1,2,4,8,13,21,31,45,66,81,97,…} be the greedy Sidon sequence: we begin with 1 and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to a+b=c+d). What is the order of growth of A? Is it true that…
Combinatorics
Erdős 340
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- Erdős and Graham [ErGr80] also asked about the difference set A−A and whether this has positive density. [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
Combinatorics
Erdős 340 - sub hasPosDensity
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- Let f(n) be the maximal k such that there exist integers 1≤a1<…<ak≤n such that all sums of the shape ∑u≤i≤vai are distinct. Is f(n)=o(n)?
Number theory
Erdős 357 - part i
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- Suppose A is an infinite set such that all finite sums of consecutive terms of A are distinct. Then it is conjectured that A has density 0.
Number theory
Erdős 357 - infinite set density
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- Suppose A is an infinite set such that all finite sums of consecutive terms of A are distinct. Then it is conjectured that the sum ∑kak1 converges.
Number theory
Erdős 357 - infinite set sum
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- Let h(n) be the maximal k such that there exist integers 1≤a1≤…≤ak≤n such that all sums of the shape ∑u≤i≤vai are distinct. Is h(n)=o(n)?
Number theory
Erdős 357 - monotone parts i
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- Let a1<a2<⋯ be an infinite sequence of integers such that a1=1 and ai+1 is the least integer which is not a sum of consecutive earlier ajs. Show that ak/k→∞.
Number theory
Erdős 359 - part i
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- Let a1<a2<⋯ be an infinite sequence of integers such that a1=1 and ai+1 is the least integer which is not a sum of consecutive earlier ajs. Show that ak/k1+c→0 for any c>0.
Number theory
Erdős 359 - part ii
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- Suppose monotone sequence A satisfies the following:
Number theory
Erdős 359 - isGoodFor 1 asymptotic
A 0 = 1and for allj,A (j + 1)is the smallest natural number that cannot be written as a sum of consecutive terms ofA 0, ..., A j. Then it is conjectured that ak loglogkklogk.- Attempts
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- There is no consecutive triple of powerful numbers.
Number theory
Erdős 364
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- Erdős [Er76d] conjectured a stronger statement: if nk is the kth powerful number, then nk+2−nk>nkc for some constant c>0. [Er76d] Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical…
Number theory
Erdős 364 - strong
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- Let P(n) denote the largest prime factor of n. Show that the set of n with P(n+1)>P(n) has density 21.
Number theory
Erdős 371
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- Show that the equation
Number theory
Erdős 373
n!=a_1!a_2!···a_k!, withn−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k, has only finitely many solutions.- Attempts
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- Hickerson conjectured the largest solution the equation
Number theory
Erdős 373 - maximal solution
n!=a_1!a_2!···a_k!, withn−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k, is16!=14!5!2!.- Attempts
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- Surányi was the first to conjecture that the only non-trivial solution to
Number theory
Erdős 373 - suranyi
a!b!=n!is6!7!=10!.- Attempts
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Number theory
Erdős 375
Erdos375Proptrue?- Attempts
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- Are there infinitely many n such that (n2n) is coprime to 105?
Number theory
Erdős 376
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- Let F(n) := \max\{m + p(m) \mid \textrm{m < ncomposite}\}\} where p(m) is the least prime divisor of m. Is it true that F(n)>n for all sufficiently large n?
Number theory
Erdős 385 - part i
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- Is it true that for every k there exists n such that ∏0≤i≤k(n−i)∣(n2n)?
Number theory
Erdős 396
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- If we only allow the digits 1 and 2 then 215 seems to be the largest such power of 2.
Number theory
Erdős 406 - one two
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- If n>1 then the iteration n↦σ(n)−1 necessarily reaches a prime. Note: this is open — it is not clear that the σ iteration always terminates, since it is non-decreasing (unlike the φ iteration which is strictly decreasing).
Number theory
Erdős 409 - sigma termination
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- Let h1(n)=h(n) and hk(n)=h(hk−1(n)). Is it true, for any m,n, there exist i and j such that hi(m)=hj(n)?
Number theory
Erdős 414
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- Let
Number theory
Erdős 416 - part i
V(x)count the number ofn≤xsuch thatϕ(m)=nis solvable. DoesV(2x)/V(x)→2?- Attempts
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- Are there infinitely many primes p such that p−1 is the only n for which mn=p?
Number theory
Erdős 456 - part iii
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- Probably there is no such A for the polynomial Xk for any k≥2. This is asked in [Sek59].
Field theory and polynomials
Erdős 477 - monomial
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- Probably there is no such A for the polynomial X3.
Field theory and polynomials
Erdős 477 - X pow three
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- Is it true that, for all k=1, there are infinitely many n such that 2n≡k(modn)?
Number theory
Erdős 479
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- Let r≥3, and let fr(N) denote the size of the largest subset of {1,…,N} such that no subset of size r has the same pairwise greatest common divisor between all elements. Erdős [Er64] proved that f3(N)>Nc/loglogN for some constant c>0, and conjectured this should also be an upper…
Combinatorics
Erdős 535
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- The first open case of Erdős Problem 535 is r=3: there should exist c>0 such that f3(N)≤Nc/loglogN for all sufficiently large N.
Combinatorics
Erdős 535 - first open case
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- **Erdős Problem 567 (Q3)** Is Q3 (the 3-dimensional hypercube) Ramsey size linear?
Combinatorics
Erdős 567 - part i
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- Let r≥3. If the edges of Kr2+1 are r-coloured then there exist r+1 vertices with at least one colour missing on the edges of the induced Kr+1. In other words, there is no balanced colouring. A conjecture of Erdős and Gyárfás [ErGy99].
Combinatorics
Erdős 617
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- Let X be a finite set of size n and H(n) be such that there is a function f:{A:A⊆X}→X so that for every Y⊆X with ∣Y∣≥H(n) we have {f(A):A⊆Y}=X. Prove that H(n)−log2n→∞.
Combinatorics
Erdős 624
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- Can the product of an arithmetic progression of positive integers n,n+d,...,n+(k−1)d of length ≥ 4, with (n,d)=1, be a perfect power?
Number theory
Erdős 672
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- Denote by M(n,k) the least common multiple of the finite set {n+1,…,n+k}. Is it true that for all m≥n+k, we get M(m,k)=M(n,k)?
Number theory
Erdős 677
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- Is it true that, for all sufficiently large n, there exists some k such that p(n+k)>k2+1, where p(m) denotes the least prime factor of m?
Number theory
Erdős 680 - part i
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- Can 4 be written as 4=∏1≤i≤k(n+i)∏1≤i≤k(m+i) for some k≥2 and m≥n+k?
Number theory
Erdős 686 - four
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- In particular, is it true that ϵn=o(1)?
Number theory
Erdős 688 - part ii
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- It is open whether there exists a projective plane of order 12.
Combinatorics
Erdős 723 - eq 12
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- It is open even for k=2. Let k=2. Does ((n+k)!)2∣(2n)! hold for infinitely many n?
Number theory
Erdős 727 - k 2
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- It is probably true that
Number theory
Erdős 770 - three
h n = 3for infinitely manyn.- Attempts
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- A Conjecture of Marian Deaconescu, see p.120 in https://doi.org/10.2307/2975810 [Needed to index shift in order to avoid trivial case n=0, where the conjecture is trivially false.]
Number theory
Erdős 779
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- Is it true that, for every ϵ>0, there exist infinitely many n such that g(n)>n1−ϵ?
Number theory
Erdős 821
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- When n>1, Lehmer conjectured that ϕ(n)∣n−1 if and only if n is prime.
Number theory
Erdős 828 - lehmer conjecture
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- Does there exist a k>2 such that the k-sized subsets of {1,...,2k} can be coloured with k+1 colours such that for every A⊂{1,…,2k} with ∣A∣=k+1 all k+1 colours appear among the k-sized subsets of A?
Combinatorics
Erdős 835
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- Is it true that, for every integer t≥1, there is some integer a such that (kn)=a with 1≤k≤2n has exactly t solutions?
Number theory
Erdős 849
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- Let dn=pn+1−pn, where pn is the nth prime. Let r(x) be the smallest even integer t such that dn=t has no solutions for n≤x. Is it true that r(x)→∞?
Number theory
Erdős 853 - part i
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- Let dn=pn+1−pn, where pn is the nth prime. Let r(x) be the smallest even integer t such that dn=t has no solutions for n≤x. Is it true that r(x)/logx→∞?
Number theory
Erdős 853 - part ii
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- The density of the divisor sum set is asymptotically equivalent to c1/log(t)c2.
Number theory
Erdős 859
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- Is there an absolute constant K such that, for every C>0, if n is sufficiently large then n has at most K divisors in (n21,n21+Cn41).
Number theory
Erdős 887 - part ii
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- Let v(n,k) count the prime factors of n+k which do not divide n+i for 0≤i<k. Is it true that v0(n)=maxk≥0v(n,k)→∞ as n→∞?
Number theory
Erdős 889
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- Let vl(n)=maxk≥lv(n,k). For every fixed l, vl(n)→∞ as n→∞ [ErSe67] Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430.
Number theory
Erdős 889 - general
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- Prove that there exists some c>0 such that h(n)∼c(lognn)1/2 as n→∞.
Number theory
Erdős 912
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- A heuristic of Tao using the Cramér model for the primes suggests this is true with c=2π.
Number theory
Erdős 912 - tao
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- It is likely that there are infinitely many primes p such that 8p2−1 is also prime.
Number theory
Erdős 913 - infinite many 8p sq sub one primes
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- Let pk denote the kth prime. For infinitely many r there are at least two integers pr<n<pr+1 all of whose prime factors are <pr+1−pr.
Number theory
Erdős 932
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- 1 α
- Is 2n+1 powerful for finitely many n?
Number theory
Erdős 936 - two pow add one
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- If r≥4 then can the sum of r−2 coprime r-powerful numbers ever be itself r-powerful?
Number theory
Erdős 939
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Is it true that F(x)≤(logx)O(1)?
Number theory
Erdős 945
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Is it true that liminff(n)=1?
Number theory
Erdős 950 - part i
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Is there an infinite sequence of distinct Gaussian primes x1,x2,… such that ∣xn+1−xn∣≪1?
Number theory
Erdős 952
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- **Erdős problem 972.** Let α>1 be irrational. Are there infinitely many primes p such that ⌊pα⌋ is also prime?
Number theory
Erdős 972
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Does
Number theory
Erdős 978 - part iii
n ^ 4 + 2represent infinitely many squarefree numbers?- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- If n distinct points in R2 form a convex polygon then some vertex has at least ⌊2n⌋ different distances to other vertices.
Convex and discrete geometry
Erdős 982
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Is it true that, for every prime p, there is a prime q≤p which is a primitive root modulo p?
Number theory
Erdős 985
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Are there infinitely many solutions to ϕ(n)=ϕ(n+1), where ϕ is the Euler totient function?
Number theory
Erdős 1003
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Let t>1 be a rational number. Is ∑n=1∞tn−11=∑n=1∞tnτ(n) irrational, where τ(n) counts the divisors of n? A conjecture of Chowla.
Number theory
Erdős 1049
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- A prime p is in class 1 if the only prime divisors of p+1 are 2 or 3. In general, a prime p is in class r if every prime factor of p+1 is in some class ≤r−1, with equality for at least one prime factor. If pr is the least prime in class r, then how does pr1/r behave? Erdos conjectured…
Number theory
Erdős 1055 - erdos limit
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- A prime p is in class 1 if the only prime divisors of p+1 are 2 or 3. In general, a prime p is in class r if every prime factor of p+1 is in some class ≤r−1, with equality for at least one prime factor. If pr is the least prime in class r, then how does pr1/r behave? Selfridge…
Number theory
Erdős 1055 - selfridge limit
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Are there infinitely many primes p such that p−k! is composite for each k such that 1≤k!<p?
Number theory
Erdős 1059
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Part (ii) of Erdős Problem 1060: bound on the number of k≤n with kσ1(k)=n.
Number theory
Erdős 1060 - part ii
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Is it true that there are infinitely many p for which f(p)=p−1?
Number theory
Erdős 1072 - part i
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Regarding the first question, Hardy and Subbarao computed all EHS numbers up to 210, and write "...if this trend conditions we expect [the limit] to be around 0.5, if it exists."
Number theory
Erdős 1074 - EHSNumbers one half
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Are there only finitely many binomial coefficients with deficiency > 1?
Combinatorics
Erdős 1093 - part ii
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- For all n≥2k the least prime factor of (kn) is ≤max(n/k,k), with only finitely many exceptions.
Number theory
Erdős 1094
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Sorenson, Sorenson, and Webster [SSWE20] give heuristic evidence that logg(k)≍logkk.
Number theory
Erdős 1095 - log isTheta
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Erdős, Lacampagne, and Selfridge [ELS93] write 'it is clear to every right-thinking person' that g(k)≥exp(clogkk) for some constant c>0.
Number theory
Erdős 1095 - lower conjecture
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Let r≥2. Is every large integer the sum of at most r+1 many r-powerful numbers?
Number theory
Erdős 1107
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- For each k≥2, does the set A={∑n∈Sn!:S⊂N finite} of all finite sums of distinct factorials contain only finitely many k-th powers?
Number theory
Erdős 1108 - part i
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- The Collatz conjecture states that for any positive integer n, there exists a natural number m such that the m-th term of the sequence is 1.
Number theory
Erdős 1135
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Are there infinitely many n>2 such that n−2k is prime for all k≥1 with 2k<n? The only known such n are 4,7,15,21,45,75,105 (OEIS [A039669](https://oeis.org/A039669)).
Number theory
Erdős 1142
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Prove that F(n)→∞ as n→∞.
Number theory
Erdős 1203
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Are there n such that n+22k is infinitely often squarefree?
Number theory
Erdős 1209 - part iii d
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Let G be an abelian group of size N, and suppose that A⊂G has density α. Are there at least α15N10 tuples (x1,…,x5,y1,…,y5)∈G10 such that xi+yj∈A whenever j∈{i,i+1,i+2}? Note: We interpret indices modulo 5.
Combinatorics
Green12.green_12
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Does there exist a Lipschitz function f:N→Z whose graph Γ={(n,f(n)):n∈N}⊆Z2 is free of 3-term progressions?
Combinatorics
Green15.green_15
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Suppose that G is a finite group, and let A⊂G×G be a subset of density α. Is it true that there are ≫α∣G∣3 triples x,y,g such that (x,y),(gx,y),(x,gy) all lie in A? Note: A is taken as α-dense, i.e. ∣A∣≥α∣G∣2 [Au16, Question 2]
Combinatorics
Green18.green_18
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Conjecture p.579 in [Aa19]: (13+o(1))n2.
Combinatorics
Green24.variants.conjecture
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Suppose that A is a K-approximate group (not necessarily abelian). Is there S⊂A, ∣S∣≫K−O(1)∣A∣, with S8⊂A4?
Group theory
Green29.green_29
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Suppose that A⊂[0,1] is open and has measure greater than 31. Is there a solution to xy=z with x,y,z∈A?
Number theory
Green3.green_3
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Let p be a prime and let A⊂Z/pZ be a set of size ⌊p⌋. Is there a dilate of A containing a gap of length 100p?
Combinatorics
Green32.green_32
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Are there infinitely many q for which there is a set A⊂Z/qZ, ∣A∣=(2+o(1))q1/2, with A+A=Z/qZ? [Gr24]
Combinatorics
Green33.green_33
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Does f(r)→∞? [Gr24]
Combinatorics
Green40.green_40
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- It is not known whether f(2) = 1 [Gr24]
Combinatorics
Green40.green_40.f_two_eq_one
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Does f~(r)→∞? [Gr24]
Combinatorics
Green40.green_40.variants.arbitrary_subsets
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Is ε−C rotations enough?
Geometry
Green41.green_41.variants.polynomial_bound
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings in 2 dimensions?
Geometry
Green42.green_42
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Let A⊂F2n be a set of density α>0. Does 10A contain a coset of some subspace of dimension at least n−O(log(1/α))? More precisely: does there exist an absolute constant C>0 such that for all n≥1 and all nonempty A⊆F2n with density α>0…
Combinatorics
Green50.green_50
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Suppose that A⊂F2n has density α>1/2−C/n. Does A+A contain a subspace of co-dimension OC(1)? [Sa11, Question 5.1]
Combinatorics
Green51.green_51.one_half
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Let K⊂Rn be a balanced compact set (that is, λK⊆K whenever ∣λ∣≤1) and suppose that the normalised Gaussian measure γn(K)≥0.99. Does 10K contain a compact convex set C with γn(C)≥0.01?
Functional analysis
Green54.green_54
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Is there an absolute constant c>0 such that, whenever A⊆N is a set of squares with ∣A∣≥2, the sumset A+A satisfies ∣A+A∣≥∣A∣1+c?
Number theory
Green60.green_60
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Does Ulam's sequence have positive density?
Number theory
Green7.green_7.variants.positive_density
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- **Green's Open Problem 72 / No-three-in-line problem**: The no-k-in-line conjecture holds for k=3.
Combinatorics
Green72.green_72
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Given n points in the unit disc, must there be a triangle of area at most n−2+o(1) determined by them?
Combinatorics
Green77.green_77
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Problem 9 (ii): is r5(N)≪N(logN)−c?
Combinatorics
Green9.green_9_ii
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α
- Problem 9 (iii): is r4(F5n)≪N1−c, where N=5n?
Combinatorics
Green9.green_9_iii
- Attempts
- 0
- Modes
- 2
- Bounty
- 1 α