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.
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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.
Fields All fields (174) Combinatorics (37) Number theory (110) Field theory and polynomials (2) Group theory (3) Measure and integration (1) Dynamical systems (1) Sequences and series (1) Harmonic analysis (1) Functional analysis (1) Geometry (2) Convex and discrete geometry (11) Probability (1) Information and communication (3) ▾
is not the sum of a prime and at most
powers of
, and pointed out that parity considerations, coupled with the fact that there are many integers not the sum of a prime and
powers of
suggest that there exist infinitely many even integers which are not the sum…
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Modes 2
Bounty $2,827 Let A = { 1 , 2 , 4 , 8 , 13 , 21 , 31 , 45 , 66 , 81 , 97 , … } A = \{1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, \ldots\} A = { 1 , 2 , 4 , 8 , 13 , 21 , 31 , 45 , 66 , 81 , 97 , … } be the greedy Sidon sequence: we begin with
for some
a , b ∈ R a,b \in \mathbb{R} a , b ∈ R and
?
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Bounty $2,827 . Is it true that
1 t ∑ 1 ≤ i < t ( s i + 1 − s i ) 2 → ∞ \frac{1}{t}\sum_{1\leq i<t}(s_{i+1}-s_i)^2 \to \infty t 1 ∑ 1 ≤ i < t ( s i + 1 − s i ) 2 → ∞
as
∣ A ∣ → ∞ \lvert A\rvert\to \infty ∣ A ∣ → ∞ ?
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Bounty $2,827 and
. Show that
f ( n ) = o ( log n ) f(n)=o(\log n) f ( n ) = o ( log n ) .
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Bounty $2,827
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Bounty $2,827
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Bounty $2,827 such that
and repeat with
replaced by
. If this terminates after finitely many steps then this produces a representation of
as the sum…
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Bounty $2,827 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 a + b = c + d a + b = c + d ). What is the order of growth of
? Is it true that…
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Bounty $2,827 such that no subset of size
has the same pairwise greatest common divisor between all elements. Erdős [Er64] proved that
f 3 ( N ) > N c / log log N f_3(N) > N^{c/\log\log N} f 3 ( N ) > N c / l o g l o g N for some constant
, and conjectured this should also be an upper…
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Bounty $2,827 for all sufficiently large
.
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Bounty $2,827 -coloured then there exist
vertices with at
least one colour missing on the edges of the induced
.
In other words, there is no balanced colouring.
A conjecture of Erdős and Gyárfás [ErGy99].
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Modes 2
Bounty $2,827 so that for every
with
∣ Y ∣ ≥ H ( n ) \lvert Y\rvert \geq H(n) ∣ Y ∣ ≥ H ( n )
we have
{ f ( A ) : A ⊆ Y } = X \left\{ f(A) : A\subseteq Y\right\}=X { f ( A ) : A ⊆ Y } = X .
Prove that
H ( n ) − log 2 n → ∞ H(n)-\log_2 n \to \infty H ( n ) − log 2 n → ∞ .
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Bounty $2,827 with
∣ A ∣ = k + 1 \lvert A\rvert=k+1 ∣ A ∣ = k + 1 all
colours appear among the
-sized subsets of
?
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Bounty $2,827 tuples
( x 1 , … , x 5 , y 1 , … , y 5 ) ∈ G 10 (x_1, \dots, x_5, y_1, \dots, y_5) \in G^{10} ( x 1 , … , x 5 , y 1 , … , y 5 ) ∈ G 10
such that
x i + y j ∈ A x_i + y_j \in A x i + y j ∈ A whenever
j ∈ { i , i + 1 , i + 2 } j \in \{i, i+1, i+2\} j ∈ { i , i + 1 , i + 2 } ?
Note: We interpret indices modulo 5.
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Bounty $2,827 is free of 3-term progressions?
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Bounty $2,827 triples
such that
( x , y ) , ( g x , y ) , ( x , g y ) (x, y), (gx, y), (x, gy) ( x , y ) , ( g x , y ) , ( x , g y )
all lie in
?
Note: A is taken as
-dense, i.e.
∣ A ∣ ≥ α ∣ G ∣ 2 |A| \ge \alpha |G|^2 ∣ A ∣ ≥ α ∣ G ∣ 2 [Au16, Question 2]
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Bounty $2,827 .
Is there a dilate of
containing a gap of length
?
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Bounty $2,827 , with
A + A = Z / q Z A + A = \mathbb{Z}/q\mathbb{Z} A + A = Z / q Z ? [Gr24]
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Bounty $2,827 contain a coset of some subspace of dimension at least
n − O ( log ( 1 / α ) ) n - O(\log(1/\alpha)) n − O ( log ( 1/ α )) ? More precisely: does there exist an absolute constant
such that for all
and all nonempty
A ⊆ F 2 n A \subseteq \mathbb{F}_2^n A ⊆ F 2 n with density
…
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Bounty $2,827 .
Does
contain a subspace of co-dimension
? [Sa11, Question 5.1]
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Bounty $2,827 , where
?
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Bounty $2,827
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