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.
Catalog
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.
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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. Is it true that
t1∑1≤i<t(si+1−si)2→∞
as
∣A∣→∞
?
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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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.
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and
k≥0
. Show that
f(n)=o(logn)
.
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.
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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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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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and
an
by
∑1≤k≤nk1=Lnan
.
Is it true that
(an,Ln)=1
occurs for infinitely many
n
?
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with
a<b
nonnegative integers are distinct.
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th powers,
is it true that
fk,3(x)≫x(3/k)
?
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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…
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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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has density
21
.
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?
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is
p
?
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p(m)
is the least
prime divisor of
m
. Is it true that
F(n)>n
for all sufficiently large
n
?
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. Is it true, for any
m,n
, there exist
i
and
j
such that
hi(m)=hj(n)
?
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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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?
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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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has
size at least
ϵN
then there must be distinct
a,b,c∈A
such that
[a,b]=[b,c]=[a,c],
where
[⋅,⋅]
denotes the least common multiple?
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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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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].
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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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, be a perfect power?
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.
Is it true that for all
m≥n+k
, 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 some
k≥2
and
m≥n+k
?
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. Is it
true that
limk→∞qk1/k=∞?
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?
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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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hold for infinitely many n?
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?
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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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.
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divisors in
(n21,n21+Cn41)
.
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for
0≤i<k
. Is it true that
v0(n)=maxk≥0v(n,k)→∞
as
n→∞
?
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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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as
n→∞
.
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all of whose prime factors are
<pr+1−pr
.
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?
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different distances to other vertices.
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irrational, where
τ(n)
counts the divisors of
n
?
A conjecture of Chowla.
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?
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.
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, with only
finitely many exceptions.
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.
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for some constant
c>0
.
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of all finite sums of distinct factorials contain only finitely many
k
-th powers?
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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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such that the restricted sumset
S+^S
is disjoint from
A
?
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?
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, where
N=5n
?
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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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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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.
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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10A
contain a coset of some subspace of dimension at least
n−O(log(1/α))
? More precisely: does there exist an absolute constant