Every problem here was open when it entered the pool.
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.
Open problems · Conjectures.io
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Every problem here was open when it entered the pool.
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.
and every sufficiently large integer can be written as
p+a
for some prime
p
and
a∈A
?
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for all
ε>0
?
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(aside from the trivial coincidences). Is it true that
liminfN→∞N1/3∣A∩{1,…,N}∣=0?
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is the natural density of
{n:φ(n)<cn}
. Is it true that there is no
x
such
that the derivative
f′(x)
exists and is positive?
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is the smallest such integer, then
ana→∞
as
a→∞
?
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in the above formulation.
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exists and is
=0
?
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irrational?
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for every finite
n≥2
(see
erdos_rado
), which covers all red ordinals below
ω⋅2=ω+ω
. This variant asks whether the result extends to…
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many distinct distances.
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n
there are at least two
(and probably many) such
A
which are non-similar.
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apart.
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distinct distances. Does
h(n)/n→∞
?
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points which form the vertices of a convex
n
-gon.
Prove that
f(n)=2n−2+1
.
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for some
a,b∈R
and
a=0
?
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be integers of gcd equal to
1
such that
∑1≤i≤rdi−11≥1.
Can all sufficiently large integers be written as a sum of the shape
∑iciai
where
ci∈{0,1}
and
ai
is divisible by
dik
and has only the digits
0,1
…
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and yet
p2
does not divide the right hand side. [Er82c] Erdős, Paul, "Miscellaneous problems in number theory".…
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, where
rk(N)
the largest possible size of a subset
of
{1,…,N}
that does not contain any non-trivial
k
-term arithmetic progression.
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,
limx→∞x1∑sn≤x(sn+1−sn)α
exists?
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. Is it true that
t1∑1≤i<t(si+1−si)2→∞
as
∣A∣→∞
?
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for all sufficiently large
N
?
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, but he is 'very doubtful'.
[Er79] Erdős, Paul, __Some unconventional problems in number theory__. Math. Mag. (1979), 67-70.
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and
k≥0
. Show that
f(n)=o(logn)
.
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such that
na=x1+y1+z1.
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and
∑an1∈Q
.
Then, for all sufficiently large
n≥1
,
an=an−12−an−1+1
.
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and
k≥0
, have density
>0
?
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Is
∑k=1∞2nk1
transcendental?
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irrational? Here
ϕ
is the Euler totient function.
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irrational? Here
pn
is the
n
-th prime (
p1=2,p2=3,…
).
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irrational?
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be the set of positive integers whose prime factors
are all in
P
. Is the sum
∑n=1∞[a1,…,an]1
irrational?
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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.
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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…
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such that
∑i=1kni1=1
,
we must have
max(ni+1−ni)≥3
?
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and
an
by
∑1≤k≤nk1=Lnan
.
Is it true that
(an,Ln)=1
occurs for infinitely many
n
?
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?
Asked by Barbeau [Ba76].
[Ba76] Barbeau, E. J., _Computer challenge corner: Problem 477: A brute force program._
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whenever the left-hand side is not zero?
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for all
ϵ>0
?
This would have significant applications to Waring's problem. Erdős and Graham describe this as
'unattackable by the methods at our disposal'.
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for sufficiently large
x
?
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.
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nonnegative integers are distinct.
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fk,3(x)≫x(3/k)
?
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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…
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complete?
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such that all sums of the shape
∑u≤i≤vai
are distinct. Is
f(n)=o(n)
?
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converges.
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such that all sums of the shape
∑u≤i≤vai
are distinct. Is
h(n)=o(n)
?
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and
ai+1
is the
least integer which is not a sum of consecutive earlier
aj
s. Show that
ak/k→∞
.
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and
ai+1
is the
least integer which is not a sum of consecutive earlier
aj
s. Show that
ak/k1+c→0
for any
c>0
.
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.
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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…
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2
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has density
21
.
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is
p
?
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is the least
prime divisor of
m
. Is it true that
F(n)>n
for all sufficiently large
n
?
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0 α
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for some constant
ck
?
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0 α
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2
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0
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2
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. Is it true that
limk→∞σk(n)k1=∞
? This is problem (iii) from Erdos, Granville, Pomerance, Spiro "On the normal behavior of the iterates of some arithmetical functions" (page 169 of the book "Analytic Number Theory"…
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.
Is it true that, for every
m,n≥2
, there exist some
i,j
such that
σi(m)=σj(n)
?
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0 α
. Is it true, for any
m,n
, there exist
i
and
j
such that
hi(m)=hj(n)
?
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with
0<a<n
and
liminfπ(x)∣A∩[1,x]∣>0?
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such that
ab≡1(modp)
?
This is discussed in this MathOverflow question [MathOverflow].
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?
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be the
k
-th prime.
Is it true that for all
k≥1
,
lcm(1,…,pk+1−1)<pk⋅lcm(1,…,pk)
?
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, there is a composite number
m
such that
n+f(n)<m<n+p(m)
Here
p(m)
is the least prime factor of
m
.
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?
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? This is also known as the Littlewood conjecture.
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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…
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for all sufficiently large
N
.
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2
-coloured then there is a monochromatic copy of the complete
3
-uniform
hypergraph on
n
vertices.
Is there some constant
c>0
such that
R3(n)≥22cn?
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(the octahedron) and at least
δn2
edges, must
G
contain an independent set of size
≫δn
? This is a problem of Erdős, Hajnal, Sós, and Szemerédi [EHSS83]. It is **open**; they proved the statement…
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as
n→∞
?
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-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].
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so that for every
Y⊆X
with
∣Y∣≥H(n)
we have
{f(A):A⊆Y}=X
.
Prove that
H(n)−log2n→∞
.
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and let
R(xi)=#{∣xj−xi∣:j=i}
,
where the points are ordered such that
R(x1)≤⋯≤R(xn).
Let
g(n)
be the maximum number of distinct values the
R(xi)
can take. Is it true that
g(n)≥(1−o(1))n
?
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, be a perfect power?
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, we get
M(m,k)=M(n,k)
?
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where
p(m)
denotes the least prime factor of
m
?
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, for all
ϵ>0
, where
Cϵ>0
is some constant?
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0
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0 α
,
where
p(m)
is the least prime factor of
m
?
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0
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2
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0 α
for some
k≥2
and
m≥n+k
?
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0
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2
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0 α
for some
k≥2
and
m≥n+k
?
Attempts
0
Modes
2
Bounty
0 α
. Is it
true that
limk→∞qk1/k=∞?
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and
q(k)≤exp(k(logk)1+o(1))?
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2
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0 α
?
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x
such that whenever
F′⊆F
is an intersecting subfamily we have…
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?
A conjecture of Erdős, Graham, Ruzsa, and Straus [EGRS75].
By
n∈(p/2,p)(modp)
we mean
n≡r(modp)
for some integer
r
with
p/2<r<p
.
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1
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-
and yet
1A∗1A(n)≪ϵ1
for all
n
?
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with
∣B∣≥h(n)
such that if
a1+⋯+ar=b1+⋯+bs
with
ai,bi∈B
then
r=s
.
Is
h(n)=Θ(n)
?
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c>0
, for all large
n
?
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?
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0 α
0
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0
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? That is, does there exist a natural
number
C
such that the number of representations of
n
as a sum of two cubes is
O((logn)C)
as
n→∞
?
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with
∣A∣=k+1
all
k+1
colours appear among the
k
-sized subsets of
A
?
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with
1≤k≤2n
has exactly
t
solutions?
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, where
pn
is the
n
th 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)→∞
?
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, where
pn
is the
n
th 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→∞
?
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0 α
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and let
F(A,X,k)
count the number of
i
such that
[ai,ai+1,…,ai+k−1]<X
, where the left-hand side is the least common
multiple. Is it true that, for every
ϵ>0
, there exists some
k
such that
F(A,X,k)<Xϵ
?
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such that
∣∩iD(Ni)∣≥k
?
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is
Oϵ(1)
?
Erdős attributes this conjecture to Ruzsa.
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divisors in
(n21,n21+Cn41)
.
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. Is it true that
v0(n)=maxk≥0v(n,k)→∞
as
n→∞
?
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0
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. 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.
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,
liminfn→∞∑0≤i<kωk(n+i)≤k?
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where
ω
counts the number of distinct prime factors without restriction?
Attempts
0
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2
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0 α
. Is it true that, for all sufficiently large
n
, there must exist an integer in
[n,n+p1⋯pk)
with
>k
many prime factors?
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tend to infinity?
(Other finite limits have been ruled out by [KoLu25], see below)
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0 α
as
n→∞
.
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0 α
are disjoint intervals of consecutive integers,
all of length at least
k
, then
∏1≤i≤r∏m∈Iim
is not a perfect power?
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such that
∏1≤i≤k1(n1+i)and∏1≤j≤k2(n2+j)
have the same prime factors?
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all of whose prime factors are
<pr+1−pr
.
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, then is it true that
limsupn→∞nlogn2k3l=∞
?
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and, for infinitely many
n
,
h(n)>(logn)c−o(1)
.
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for every
n
?
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>r
?
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of cardinality continuum such that
A+A⊆R∖S
?
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0 α
?
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such that
∑n≤xτ(f(n))≈c⋅xlogx
? Note that it is unclear whether the polynomial should have integer coefficients or merely be integer-valued. We…
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there exists
n
such that
pk−2∤f(n)
,
then are there infinitely many
n
for which
f(n)
is
(k−2)
-power-free?
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,
where the
pi
are prime numbers. Is it true that
limsupfk(n)=∞
?
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different distances to other vertices.
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0 α
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irrational, where
τ(n)
counts the divisors of
n
?
A conjecture of Chowla.
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such that
n∈Ii∏n≡1modn
for all
1≤i≤k
?
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0 α
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0 α
0 α
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distinct distances?
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, with only
finitely many exceptions.
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0 α
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0 α
for some constant
c>0
.
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0
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2
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0 α
,
F(n)>n
for sufficiently large
n
.
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0 α
2
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of all finite sums of distinct factorials contain only finitely many
k
-th powers?
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0 α
, where
pn
denotes the
n
th prime. Is it true that
(maxn<xdn)2maxn<xdndn−1→0
as
x→∞
?
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0 α
with
2k<n
?
The only known such
n
are
4,7,15,21,45,75,105
(OEIS [A039669](https://oeis.org/A039669)).
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0 α
(see…
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0
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2
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0 α
for all
large
n
) such that
∑n≤xfr(n)2≪x
for all
x
?
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0
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2
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0 α
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2
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such that the restricted sumset
S+^S
is disjoint from
A
?
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0
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2
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0 α
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0
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2
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0 α
, where
N=5n
?
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0
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0 α
tuples
(x1,…,x5,y1,…,y5)∈G10
such that
xi+yj∈A
whenever
j∈{i,i+1,i+2}
?
Note: We interpret indices modulo 5.
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is free of 3-term progressions?
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1
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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]
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.
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.
Is there a dilate of
A
containing a gap of length
100p
?
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, with
A+A=Z/qZ
? [Gr24]
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, can we almost surely cover
Z/pZ
with
100p
translates of
A
? [Gr24]
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for all sufficiently large
p
. Is it true…
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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
…
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.
Does
A+A
contain a subspace of co-dimension
OC(1)
? [Sa11, Question 5.1]
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contain a composite number?
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satisfies
∣A+A∣≥∣A∣1+c
?
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congruent to some product
a1a2
where
a1,a2∈A
?
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?
We formalize this as an eventual statement for sufficiently large real
X
.
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is optimal.
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, and
s
is not a trivial zero
−2(n+1)
for some
n∈N
, then
Re(s)=21
. This is the official Millennium Prize Problem as posed by…
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is a primitive root modulo
p
has a positive asymptotic
density inside the set of primes. In particular,
S(a)
is infinite.
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0 α
.
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satisfies Schinzel condition,
there exist infinitely many natural numbers
m
such that
fi(m)
are primes for all
i
.
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denote the number of primes
p≤n
such that
(p,p+m1,…,p+mk)
forms an admissible prime constellation. Let
w(q;m1,…,mk)
denote the number of distinct residues of
0,m1,…,mk
modulo
q
, and…
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where
π(z)
denotes the prime-counting function, giving the number of primes up to
and including