Vela Mathematics Program: problems

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80 problems · 1/2Number SourcesGraph view

declared verifiable · formalizederdos:7

Can there be a finite covering system of the integers with distinct moduli, all of which are odd and greater than

1

number theorycovering systemsN/A4 sources

declared open · formalizederdos:12

Let

A \subset \mathbb{N}

be infinite with no distinct

a, b, c \in A

such that

a \mid (b + c)

with

b, c > a

. Can

|A \cap [1, N]|/\sqrt{N}

have positive lower limit? Must every such

A

fall below

N^{1-c}

infinitely often?

number theoryN/A4 sources

declared disproved (Lean) · formalizederdos:26

Let

A\subset\mathbb{N}

be infinite. Must there exist some

k\geq 1

such that almost all integers have a divisor of the form

a+k

for some

a\in A

? The question as posed follows negatively from Davenport–Erdős (1951). The AI result settles Tenenbaum's harder variant, also negatively: there is an infinite

A

such that for every

k\geq 1

the set of multiples of

A+k

has upper density below

0.34

number theorydivisorsN/A6 sources

declared proved (Lean) · formalizederdos:38

True ↔ ∃ B, ¬B.IsWeakAddBasis ∧ ∃ f, (∀ (α : ℝ), 0 < α → α < 1 → f α > 0) ∧ ∀ (A : Set ℕ) (N : ℕ), have α := schnirelmannDensity A; ∃ b ∈ B, ↑(Set.Ioc 0 N ∩ (A ∪ (A + {b}))).ncard ≥ (α + f α) * ↑N

number theoryN/A6 sources

declared solved (Lean) · formalizederdos:42

True ↔ ∀ M ≥ 1, ∀ᶠ (N : ℕ) in Filter.atTop, ∀ (A : Set ℕ), A.IsMaximalSidonSetIn N → ∃ B ⊆ Set.Icc 1 N, IsSidon B ∧ B.ncard = M ∧ (A - A) ∩ (B - B) = {0}

number theorysidon setsadditive combinatoricsN/A6 sources

declared disproved · formalized · $100erdos:43

If Sidon sets

A, B \subseteq \{1, \dots, N\}

satisfy

(A-A) \cap (B-B) = \{0\}

, must

\binom{|A|}{2} + \binom{|B|}{2} \le \binom{f(N)}{2} + O(1)

, where

f(N)

is the largest Sidon-set size in

[N]

- and can the bound be improved by a fixed proportion when

|A| = |B|

number theorysidon setsadditive combinatoricsA143824 · A227590 · A003022 · possible4 sources

declared proved (Lean) · formalized · $250erdos:123

Let

a,b,c>1

be pairwise coprime integers. Is every large integer a sum of distinct numbers of the form

a^k b^l c^m

(

k,l,m\ge 0

), none dividing another?

number theoryN/A5 sources

declared disproved (Lean) · formalizederdos:125

False ↔ ({x | (Nat.digits 3 x).toFinset ⊆ {0, 1}} + {x | (Nat.digits 4 x).toFinset ⊆ {0, 1}}).HasPosDensity

number theorybase representationsA3670906 sources

declared openerdos:131

Let

F(N)

be the maximal size of

A\subseteq\{1,\ldots,N\}

such that no

a\in A

divides the sum of any nonempty subset of

A\setminus\{a\}

. Estimate

F(N)

. The lower bound

F(N)\gg N^{1/5}

is classical, from constructions of Erdős and Csaba, and every non-dividing set is non-averaging, which gave

F(N)\leq N^{1/4+o(1)}

. The claimed new result is the matching upper bound

F(N)\leq N^{1/5+o(1)}

, obtained by running the Pham-Zakharov density-increment argument one dimension lower through a projective normalization, hence

F(N)=N^{1/5+o(1)}

number theoryA0680632 sources

declared disproved (Lean) · formalizederdos:205

False ↔ ∀ᶠ (n : ℕ) in Filter.atTop, Erdos205.IsRepresentable (fun m => Real.log (Real.log ↑m)) n

number theorypossible6 sources

declared open · formalizederdos:254

A \subseteq \mathbb{N}

has unbounded dyadic-shell counts and

\sum_{n \in A} \|\theta n\| = \infty

for every

0 < \theta < 1

, must

A

be complete - is every sufficiently large integer a sum of distinct elements of

A

number theoryN/A4 sources

declared proved (Lean) · formalizederdos:281

True ↔ ∀ (n : ℕ → ℕ), StrictMono n → (∀ (i : ℕ), 0 < n i) → (∀ (a : Erdos281.ResidueChoice n), (Erdos281.avoidAll n a).HasIntDensity 0) → ∀ (ε : ℝ), 0 < ε → ∃ k, ∀ (a : Erdos281.ResidueChoice n), ∃ d, (Erdos281.avoidPrefix n a k).HasIntDensity d ∧ d < ε

number theorycovering systemsN/A7 sources

declared proved (Lean) · formalizederdos:283

True ↔ ∀ (p : Polynomial ℤ), Erdos283.Condition p

number theoryunit fractionsA3807916 sources

declared open · formalizederdos:306

a/b \in \mathbb{Q}_{>0}

and

b

is squarefree, can

a/b

always be written as a finite sum of reciprocals of distinct products of two distinct primes?

number theoryunit fractionsN/A4 sources

declared solvederdos:320

Let

S(N)

count the distinct values of

\sum_{n\in A} 1/n

over

A\subseteq\{1,\dots,N\}

. Estimate

S(N)

number theoryunit fractionsA0722072 sources

declared solved · formalizederdos:321

What is the largest

A\subseteq\{1,\dots,N\}

such that all subset sums

\sum_{n\in S}1/n

(over

S\subseteq A

) are distinct?

number theoryunit fractionsA384927 · A3915923 sources

declared open · formalizederdos:326

Does there exist

A=\{a_1<a_2<\cdots\}\subset \mathbb{N}

which is a minimal basis of order

2

(every large integer is the sum of

2

elements from

A

, and no proper subset of

A

has this property) such that

\lim_{k\to \infty}a_k/k^2=c

for some

c\neq 0

? A claimed construction gives a minimal basis with

A(x)=C\sqrt{x}+O(1)

, answering the question affirmatively; Erdős and Graham had conjectured a negative answer.

number theoryadditive basisN/A4 sources

declared proved (Lean) · formalizederdos:330

True ↔ ∃ A h, Erdos330.MinAsymptoticAddBasisOfOrder A h ∧ 0 < A.upperDensity ∧ ∀ n ∈ A, 0 < (Erdos330.UnrepWithout A n h).upperDensity

number theoryadditive basisN/A5 sources

declared disproved (Lean) · formalizederdos:333

False ↔ ∀ (A : Set ℕ), A.HasDensity 0 → ∃ B, A ⊆ B + B ∧ (fun N => ↑(B ∩ Set.Icc 1 N).ncard) =o[Filter.atTop] fun N => √↑N

number theoryadditive basisN/A6 sources

declared openerdos:336

h(r)

is the maximal finite exact order attainable by an additive basis of order at most

r

, what is

\lim_{r \to \infty} h(r)/r^2

? The candidate proof identifies the sharp limit

1/3

number theoryadditive basispossible2 sources

declared open · formalizederdos:346

Let

A=\{1\leq a_1< a_2<\cdots\}

be a set of integers such that

A\backslash B

is complete for any finite subset

B

and not complete for any infinite subset

B

. If

a_{n+1}/a_n \geq 1+\epsilon

for all

n

, must

\lim_n a_{n+1}/a_n=(1+\sqrt{5})/2

? Under the reading where the ratio limit is assumed to exist, a Lean-verified argument forces the limit to be the golden ratio; a separate construction disproves the literal statement where convergence is not assumed.

number theorycomplete sequencesN/A4 sources

declared proved (Lean) · formalizederdos:347

True ↔ ∃ a, Monotone a ∧ Filter.Tendsto (fun n => ↑(a (n + 1)) / ↑(a n)) Filter.atTop (nhds 2) ∧ ∀ (ι : ℕ → ℕ), (Set.range ι)ᶜ.Finite → (subsetSums (Set.range (a ∘ ι))).HasDensity 1

number theorycomplete sequencesN/A6 sources

declared proved (Lean) · formalizederdos:351

True ↔ ∀ (P : Polynomial ℚ), 0 < P.natDegree → 0 < P.leadingCoeff → Erdos351.HasCompleteImage P

number theorycomplete sequencesN/A6 sources

declared proved (Lean) · formalizederdos:369

True ↔ ∀ (ε : ℝ), 0 < ε → ∀ (k : ℕ), 2 ≤ k → ∀ᶠ (n : ℕ) in Filter.atTop, ∃ a, n / 2 ≤ a + 1 ∧ a + k ≤ n ∧ ∀ j < k, ∀ p ∈ (a + 1 + j).primeFactors, ↑p ≤ ↑n ^ ε

number theoryN/A7 sources

declared provederdos:380

Erdős problem 380

number theoryA070003 · A388654 · A387054 · A3891003 sources

declared open · formalizederdos:390

Let

f(n)

be the least

m

for which

n!

can be written as

a_1\cdots a_k

with

n < a_1 < \cdots < a_k = m

- the smallest possible largest factor in a factorization of

n!

into distinct integers all exceeding

n

. Erdős, Guy and Selfridge proved

f(n) - 2n \asymp n/\log n

. Erdős asked whether there is a constant

c

with

f(n) - 2n \sim c\,\frac{n}{\log n},

and what it is. This preprint answers yes and names the constant:

\lim_{n\to\infty}\frac{(f(n)-2n)\log n}{n} = \frac{4029639598}{25970038185} \approx 0.15516.

number theoryfactorialsA1934294 sources

declared open · formalizederdos:394

For the least

t_k(n)

with

n \mid t_k(n)(t_k(n)+1)\cdots(t_k(n)+k-1)

, do the conjectured logarithmic-saving and adjacent-length estimates hold on average? Both answered affirmatively, with

c = 1/2048

admissible in the

t_2

bound.

number theoryA3440054 sources

declared disproved (Lean) · formalizederdos:397

False ↔ {(M, N) | Disjoint M N ∧ ∏ i ∈ M, i.centralBinom = ∏ j ∈ N, j.centralBinom}.Finite

number theorybinomial coefficientsN/A7 sources

declared proved (Lean) · formalizederdos:401

True ↔ ∃ f, Filter.Tendsto f Filter.atTop Filter.atTop ∧ ∀ (r : ℕ), 1 ≤ r → {n | ∃ a₁ a₂, 0 < a₁ ∧ 0 < a₂ ∧ ↑a₁ + ↑a₂ > ↑n + f r * Real.log ↑n ∧ a₁.factorial * a₂.factorial ∣ n.factorial * (∏ i ∈ Finset.range r, Nat.nth Nat.Prime i) ^ n}.Infinite

number theoryfactorialsN/A6 sources

declared open · formalizederdos:424

Let

a_1 = 2

and

a_2 = 3

and continue the sequence by appending to

a_1, \dots, a_n

all possible values of

a_ia_j - 1

with

i \ne j

. Is it true that the set of integers which eventually appear has positive density?

number theoryA0052444 sources

declared open · formalizederdos:450

How large must

y(\varepsilon, n)

be so that every interval

(x, x+y)

contains at most

\varepsilon y

integers having a divisor in

(n, 2n)

? The candidate proof gives the sharp fixed-

\varepsilon

order

y = \Theta_\varepsilon(n)

, uniformly in the translate.

number theorydivisorsN/A4 sources

declared openerdos:451

Let

n_k

be the least integer greater than

2k

for which

\prod_{i=1}^k (n_k - i)

has no prime factor in

(k, 2k)

. How rapidly must

n_k

grow?

number theoryA3866203 sources

declared proved (Lean) · formalizederdos:457

True ↔ ∃ ε > 0, {n | ∀ (p : ℕ), ↑p ≤ (2 + ε) * Real.log ↑n → Nat.Prime p → p ∣ ∏ i ∈ Finset.Icc 1 ⌊Real.log ↑n⌋₊, (n + i)}.Infinite

number theoryA3916686 sources

declared proved (Lean) · formalizederdos:469

Does the sum of the reciprocals of all primitive pseudoperfect numbers converge?

number theorydivisorsA006036 · A119425 · possible5 sources

declared open · formalizederdos:477

Does there exist an integer polynomial

f

of degree at least two and a set

A \subseteq \mathbb{Z}

such that every integer has a unique representation

n = a + f(k)

? A manuscript claims the thirteenth powers admit a tiling complement.

number theoryN/A4 sources

declared open · formalizederdos:489

A

is a forbidden-divisor set with

|A \cap [1,x]| = o(\sqrt{x})

and

B = \{b_1 < b_2 < \cdots\}

the sifted set, must

x^{-1} \sum_{b_i < x} (b_{i+1} - b_i)^2

converge to a finite limit?

number theoryN/A4 sources

declared open · formalizederdos:538

If each integer has at most

r

representations

m = pa

with

p

prime and

a \in A \subseteq [1, N]

, what is the best upper bound for

\sum_{a \in A} 1/a

? The candidate proof gives the matching order

\Theta_r(\log N / \log\log N)

number theoryN/A5 sources

declared open · formalizederdos:539

For

|A| = n

, how small can the cofactor set

Q(A) = \{a / \gcd(a,b) : a, b \in A\}

be? The answer is

h(n) = n^{1/2 + o(1)}

: a new upper bound

h(n) \le n^{1/2} \exp(O(\sqrt{\log n}))

matches the classical lower bound.

number theorypossible5 sources

declared disprovederdos:543

Erdős problem 543

number theorygroup theorypossible3 sources

declared solved (Lean) · formalizederdos:650

∀ (m : ℕ), Erdos650.f m = min m ⌈2 * √↑m⌉₊

number theoryA0274346 sources

declared openerdos:684

For the least

k

at which the small-prime part of

\binom{n}{k}

exceeds

n^2

, how large can

f(n)

be?

number theoryprimesbinomial coefficientsA392019 · possible3 sources

declared solvederdos:690

Erdős problem 690

number theorypossible4 sources

declared solved (Lean) · formalizederdos:694

∀ (fmax fmin : ℕ → ℕ), (∀ (n : ℕ), IsGreatest (Nat.totient ⁻¹' {n}) (fmax n)) → (∀ (n : ℕ), IsLeast (Nat.totient ⁻¹' {n}) (fmin n)) → ∃ o, Filter.Tendsto o Filter.atTop (nhds 0) ∧ ∀ (x : ℕ), sSup {x_1 | ∃ n, ∃ (_ : n ≤ x) (_ : ∃ m, m.totient = n), ↑(fmax n) / ↑(fmin n) = x_1} = (Real.exp Real.eulerMascheroniConstant + o x) * Real.log (Real.log ↑x)

number theorypossible6 sources

declared solved (Lean)erdos:696

Erdős problem 696

number theorydivisorspossible4 sources

declared openerdos:709

How long must an interval be to contain distinct representatives

x_i

, with

a_i \mid x_i

, for every

n

-element set of moduli

A = \{a_1, \dots, a_n\}

number theorypossible2 sources

declared proved (Lean) · formalizederdos:728

Whether there are infinitely many integers

a, b, n

with

a, b \ge \varepsilon n

such that

a!\cdot b!

divides

n!\cdot(a+b-n)!

while

a+b

exceeds

n

by more than

C\cdot\log n

number theoryfactorialsN/A6 sources

declared proved (Lean) · formalizederdos:729

True ↔ ∀ C > 0, ∃ K ≥ 3, {(a, b, n) | a > 0 ∧ b > 0 ∧ n > 0 ∧ ↑a + ↑b > ↑n + C * Real.log ↑n ∧ ∀ (p : ℕ), Nat.Prime p → p > K → padicValNat p (↑n.factorial / (↑a.factorial * ↑b.factorial)).den = 0}.Infinite

number theoryfactorialsN/A6 sources

declared openerdos:731

Let

A(n)

be the least positive integer not dividing

\binom{2n}{n}

. Erdos asked for the behaviour of

A(n)

for reasonable

n

. Under an explicit dyadic-regularity formalization of reasonable, the distribution is determined on dyadic intervals against the scale

F_X = \sqrt{2}(\log 2)^{1/4} L^{1/4} \exp\sqrt{(\log 2)L}

with

L = \log(2X)

number theorybinomial coefficientsA0061974 sources

declared openerdos:768

Let

A(x)

count

n \le x

such that every prime

p \mid n

has a divisor

d > 1

n

with

d \equiv 1 \pmod p

. Erdos asked whether

A(x)/x = \exp(-(c+o(1))\sqrt{\log x}\log\log x)

. It does, with

c = 1/(2\sqrt{\log 2})

number theoryA001034 · A3522873 sources

declared open · formalizederdos:769

For the least cutoff

c(n)

after which every

k

occurs as the number of homothetic cubes in a decomposition of the unit

n

-cube, is

c(n) \gg n^n

? The Lean proof shows

c(n) = o(n^n)

along odd dimensions.

number theorygeometryA014544 · possible4 sources

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