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.
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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.
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.
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.
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irrational?
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many distinct distances.
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for some
a,b∈R
and
a=0
?
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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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. 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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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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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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nonnegative integers are distinct.
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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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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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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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?
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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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-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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, 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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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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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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. 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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. 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…
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. 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…
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?
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, with only
finitely many exceptions.
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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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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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, with
S8⊂A4
?
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with
x,y,z∈A
?
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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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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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) and suppose that the normalised Gaussian measure