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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$4,732
, 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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$4,732
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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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2
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$4,732
and
k≥0
. Show that
f(n)=o(logn)
.
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$4,732
.
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$4,732
such that
na=x1+y1+z1.
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$4,732
irrational? Here
ϕ
is the Euler totient function.
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$4,732
irrational? Here
pn
is the
n
-th prime (
p1=2,p2=3,…
).
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$4,732
irrational?
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$591
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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0
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$591
and
an
by
∑1≤k≤nk1=Lnan
.
Is it true that
(an,Ln)=1
occurs for infinitely many
n
?
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$591
nonnegative integers are distinct.
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$4,732
fk,3(x)≫x(3/k)
?
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$591
such that all sums of the shape
∑u≤i≤vai
are distinct. Is
f(n)=o(n)
?
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$4,732
converges.
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$4,732
such that all sums of the shape
∑u≤i≤vai
are distinct. Is
h(n)=o(n)
?
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2
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$4,732
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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$4,732
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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$4,732
.
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$4,732
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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$4,732
has density
21
.
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$4,732
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is
p
?
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$591
is the least
prime divisor of
m
. Is it true that
F(n)>n
for all sufficiently large
n
?
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0
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$4,732
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$4,732
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2
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$4,732
0
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$4,732
. Is it true, for any
m,n
, there exist
i
and
j
such that
hi(m)=hj(n)
?
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$4,732
such that
ab≡1(modp)
?
This is discussed in this MathOverflow question [MathOverflow].
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$591
?
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$4,732
?
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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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$4,732
for all sufficiently large
N
.
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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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$591
, be a perfect power?
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$4,732
, we get
M(m,k)=M(n,k)
?
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$4,732
where
p(m)
denotes the least prime factor of
m
?
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0
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2
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$4,732
for some
k≥2
and
m≥n+k
?
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0
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2
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$4,732
. Is it
true that
limk→∞qk1/k=∞?
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$591
?
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$591
?
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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$591
hold for infinitely many n?
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$4,732
?
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0
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with
1≤k≤2n
has exactly
t
solutions?
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$4,732
, 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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$4,732
, 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
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2
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$4,732
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$4,732
divisors in
(n21,n21+Cn41)
.
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0
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$4,732
. Is it true that
v0(n)=maxk≥0v(n,k)→∞
as
n→∞
?
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0
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$4,732
. 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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$4,732
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$4,732
$4,732
all of whose prime factors are
<pr+1−pr
.
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$4,732
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$4,732
?
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irrational, where
τ(n)
counts the divisors of
n
?
A conjecture of Chowla.
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$4,732
?
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$4,732
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$4,732
, with only
finitely many exceptions.
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$4,732
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$4,732
for some constant
c>0
.
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2
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$4,732
2
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$4,732
of all finite sums of distinct factorials contain only finitely many
k
-th powers?
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$4,732
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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$4,732
such that the restricted sumset
S+^S
is disjoint from
A
?
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$591
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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$4,732
is free of 3-term progressions?
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$4,732
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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$4,732
.
Is there a dilate of
A
containing a gap of length
100p
?
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$4,732
, with
A+A=Z/qZ
? [Gr24]
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$4,732
contain a coset of some subspace of dimension at least
n−O(log(1/α))
? More precisely: does there exist an absolute constant