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604 problems · 6/13Number SourcesGraph view

declared proved · formalizederdos:402

∀ (A : Finset ℕ), 0 ∉ A → A.Nonempty → ∃ a ∈ A, ∃ b ∈ A, ↑(a.gcd b) ≤ ↑a / ↑A.card

number theoryN/A2 sources

declared proved (Lean) · formalizederdos:403

True ↔ {p | (∀ a ∈ p.2, 0 < a) ∧ 2 ^ p.1 = ∑ a ∈ p.2, a.factorial}.Finite

number theoryfactorialsN/A4 sources

declared open · formalizederdos:406

True ↔ {n | n.isPowerOfTwo ∧ Nat.digits 3 n ⊆ [0, 1]}.Finite

number theorybase representationsN/A2 sources

declared open · formalizederdos:409

∀ (n : ℕ), 0 < n → IsLeast {i | Nat.Prime ((fun x => x.totient + 1)^[i] n)} sorry

number theoryiterated functionsA039651 · A2294872 sources

declared open · formalizederdos:410

True ↔ ∀ n > 1, Filter.Tendsto (fun k => ↑((⇑(ArithmeticFunction.sigma 1))^[k] n) ^ (1 / ↑k)) Filter.atTop Filter.atTop

number theoryiterated functionsA007497 · possible2 sources

declared open · formalizederdos:412

True ↔ ∀ m ≥ 2, ∀ n ≥ 2, ∃ i j, (⇑(ArithmeticFunction.sigma 1))^[i] m = (⇑(ArithmeticFunction.sigma 1))^[j] n

number theoryiterated functionsA007497 · A0515722 sources

declared open · formalizederdos:413

True ↔ {n | Erdos413.IsBarrier (fun m => ↑(ArithmeticFunction.cardDistinctFactors m)) n}.Infinite

number theoryiterated functionsA0052362 sources

declared open · formalizederdos:414

True ↔ ∀ m > 0, ∀ n > 0, ∃ i j, Erdos414.h^[i] m = Erdos414.h^[j] n

number theoryiterated functionsA064491 · possible2 sources

declared open · formalizederdos:416

Filter.Tendsto (fun x => Erdos416.V (2 * x) / Erdos416.V x) Filter.atTop (nhds 2)

number theoryA2648102 sources

declared open · formalizederdos:417

True ↔ ∃ L, Filter.Tendsto (fun x => ↑{k | k ∈ Set.range Nat.totient ∧ ↑k ≤ x}.ncard / ↑(Nat.totient '' {m | 1 ≤ m ∧ ↑m ≤ x}).ncard) Filter.atTop (nhds L)

number theoryA264810 · A0610702 sources

declared proved (Lean) · formalizederdos:418

True ↔ {x | ∃ n, n - n.totient = x}ᶜ.Infinite

number theoryA005278 · A2639585 sources

declared solved (Lean) · formalizederdos:419

{x | MapClusterPt x Filter.atTop Erdos419.factorialDivisorRatio} = Erdos419.limitPointSet

number theoryfactorialsN/A5 sources

declared open · formalizederdos:421

True ↔ ∃ d, StrictMono d ∧ 1 ≤ d 0 ∧ (Set.range d).HasDensity 1 ∧ Set.InjOn (fun x => match x with | (u, v) => ∏ i ∈ Finset.Icc u v, d i) {(u, v) | u ≤ v}

number theoryA389544 · A3908482 sources

declared open · formalizederdos:422

True ↔ {n | ∀ (x : ℕ+), Erdos422.f x ≠ n}.Infinite

number theoryA0051852 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 disproved (Lean) · formalized · $25erdos:426

False ↔ ∃ c, 0 < c ∧ ∃ᶠ (n : ℕ) in Filter.atTop, ∃ H, c * (2 ^ n.choose 2 / ↑n.factorial) ≤ ↑H.uniqueSubgraphCount

graph theorypossible5 sources

declared proved (Lean) · formalizederdos:427

True ↔ Erdos427.erdos427

number theoryprimesN/A5 sources

declared open · formalizederdos:428

True ↔ ∃ A, (∃ᶠ (n : ℕ) in Filter.atTop, ∀ a ∈ A, 0 < a → a < n → Nat.Prime (n - a)) ∧ Filter.liminf (fun n => Erdos428.primeDensityRatio A n) Filter.atTop > 0

number theoryprimesN/A2 sources

declared disproved (Lean) · formalizederdos:429

False ↔ ∃ f, Filter.Tendsto f Filter.atTop Filter.atTop ∧ ∀ (A : Set ℕ), A.Infinite → (∀ (N : ℕ), (A ∩ Set.Icc 1 N).ncard ≤ f N) → (∀ (p : ℕ), Nat.Prime p → ∃ b, ∀ a ∈ A, ↑a ≠ b) → ∃ n, ∀ a ∈ A, Nat.Prime (n + a)

number theoryN/A5 sources

declared proved (Lean) · formalizederdos:434

∀ (n k : ℕ), 1 ≤ n → 2 ≤ k → k ≤ n → IsGreatest {x | ∃ S, ∃ (_ : S ⊆ Finset.Icc 1 n) (_ : S.card = k) (_ : S.gcd id = 1), Erdos434.Nat.NcardUnrepresentable ↑S = x} (Erdos434.Nat.NcardUnrepresentable (Set.Icc (n - k + 1) n))

number theorypossible5 sources

declared proved (Lean) · formalizederdos:435

∀ (n : ℕ), n ≠ 0 → (∀ (p k : ℕ), Nat.Prime p → n ≠ p ^ k) → IsGreatest {m | ¬∃ c, m = ∑ i ∈ Finset.Ico 1 n, ↑(c i) * ↑(n.choose i)} (∑ p ∈ n.primeFactors, (∑ d ∈ Finset.Icc 1 (n.factorization p), ↑(n.choose (p ^ d))) * (↑p - 1) - ↑n)

number theoryA3894795 sources

declared disproved · formalizederdos:442

False ↔ ∀ (A : Set ℕ), Filter.Tendsto (fun x => 1 / Erdos442.Real.maxLogOne (Erdos442.Real.maxLogOne x) * ∑ n ∈ (A ∩ Set.Icc 1 ⌊x⌋₊).toFinset, 1 / ↑n) Filter.atTop Filter.atTop → Filter.Tendsto (fun x => 1 / (∑ n ∈ (A ∩ Set.Icc 1 ⌊x⌋₊).toFinset, 1 / ↑n) ^ 2 * ∑ nm ∈ (Erdos442.Set.bddProdUpper A x).toFinset, 1 / ↑(nm.1.lcm nm.2)) Filter.atTop Filter.atTop

number theoryN/A2 sources

declared proved (Lean) · formalizederdos:443

True ↔ ∀ (s : ℕ), ∃ m, ∃ n < m, s ≤ (Erdos443.A n ∩ Erdos443.A m).card

number theoryN/A5 sources

declared open · formalizederdos:445

True ↔ ∀ c > 1 / 2, ∀ᶠ (p : ℕ) in Filter.atTop, Nat.Prime p → ∀ (n : ℕ), Erdos445.Erdos445Prop c p n

number theoryN/A2 sources

declared proved (Lean) · formalizederdos:447

True ↔ (fun n => ↑(Erdos447.maxUnionFree n)) =o[Filter.atTop] fun n => 2 ^ n

combinatoricspossible5 sources

declared disproved · formalizederdos:448

False ↔ ∀ (ε : ℝ), 0 < ε → {n | ↑(Erdos448.tauPlus n) < ε * ↑n.divisors.card}.HasDensity 1

number theorydivisorsA3974332 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 disproved (Lean) · formalizederdos:453

False ↔ Erdos453.EventuallyHasPrimeWitness

number theoryN/A5 sources

declared open · formalizederdos:454

True ↔ Filter.limsup (fun n => ↑(Erdos454.f n) - 2 * ↑(Nat.nth Prime n)) Filter.atTop = ⊤

number theoryprimesA389676 · A3896772 sources

declared open · formalizederdos:455

True ↔ ∀ (q : ℕ → ℕ), StrictMono q → (∀ (n : ℕ), Nat.Prime (q n) ∧ q (n + 2) - q (n + 1) ≥ q (n + 1) - q n) → Filter.Tendsto (fun n => ↑(q n) / ↑n ^ 2) Filter.atTop Filter.atTop

number theoryN/A2 sources

declared open · formalizederdos:456

True ↔ Filter.Tendsto (fun N => ↑(Nat.count {n | Erdos456.m n < Erdos456.p n} N) / ↑N) Filter.atTop (nhds 1)

number theoryA034694 · A061026 · possible3 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 falsifiable · formalizederdos:458

True ↔ ∀ (k : ℕ), Erdos458.lcm_upto (Nat.nth Prime (k + 1) - 1) < Nat.nth Prime k * Erdos458.lcm_upto (Nat.nth Prime k)

number theoryprimesA0566042 sources

declared solved (Lean) · formalizederdos:459

∀ {u : ℕ}, 2 ≤ u → u + 2 ≤ Erdos459.f u ∧ Erdos459.f u ≤ u ^ 2

number theoryprimesA2892805 sources

declared open · formalizederdos:463

True ↔ ∃ f, ∃ (_ : Filter.Tendsto f Filter.atTop Filter.atTop), ∀ᶠ (n : ℕ) in Filter.atTop, ∃ m, m.Composite ∧ n + f n < m ∧ m < n + m.minFac

number theoryprimespossible2 sources

declared proved (Lean) · formalizederdos:464

True ↔ ∀ (n : ℕ → ℕ), StrictMono n → (∀ (k : ℕ), 0 < n k) → IsLacunary n → ∃ θ, Irrational θ ∧ ¬Dense (Set.range fun k => ↑(θ * ↑(n k)))

number theoryN/A4 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 · formalized · $10erdos:470

sorry ↔ ∃ n, n.Weird ∧ Odd n

number theorydivisorsA006037 · A0029752 sources

declared proved (Lean) · formalizederdos:476

True ↔ ∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (A : Finset (ZMod p)), A.restrictedSumset.card ≥ min (2 * A.card - 3) p

number theoryadditive combinatoricsN/A5 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:479

sorry ↔ ∀ k > 1, {n | 2 ^ n ≡ k [MOD n]}.Infinite

number theoryA036236 · A015919 · A050259 · A015921 · A006521 · A006517 · A0159402 sources

declared proved · formalizederdos:480

True ↔ ∀ (x : ℕ → ℝ), (∀ (n : ℕ), x n ∈ Set.Icc 0 1) → ⨅ n, Filter.liminf (fun m => ↑↑n * |x (m + ↑n) - x m|) Filter.atTop ≤ 1 / √5

number theoryN/A3 sources

declared proved (Lean) · formalizederdos:481

∀ {r : ℕ} (a b : Fin r → ℕ), (∀ (i : Fin r), 0 < a i) → 1 < ∑ i, 1 / ↑(a i) → ∃ k, ¬((Erdos481.T a b)^[k] [1]).Nodup

number theoryN/A5 sources

declared proved (Lean) · formalizederdos:484

∃ c, 0 < c ∧ ∀ (k : ℕ), 0 < k → ∃ N₀, ∀ (N : ℕ), N₀ ≤ N → ∀ (f : ℕ → Fin k), c * ↑N ≤ ↑{n ∈ Finset.Icc 1 N | ∃ a ∈ Finset.Icc 1 N, ∃ b ∈ Finset.Icc 1 N, a ≠ b ∧ f a = f b ∧ a + b = n}.card

number theoryadditive combinatoricsramsey theoryN/A5 sources

declared open · formalizederdos:486

True ↔ ∀ (X : (n : ℕ) → Set (ZMod n)), ∃ d, {m | ∀ (n : ℕ), ↑m ∉ X n}.HasLogDensity d

number theoryprimitive setsN/A2 sources

declared proved (Lean) · formalizederdos:487

True ↔ ∀ (A : Set ℕ), A.HasPosDensity → ∃ a ∈ A, ∃ b ∈ A, ∃ c ∈ A, a ≠ b ∧ b ≠ c ∧ a ≠ c ∧ a.lcm b = c

number theoryN/A5 sources

declared falsifiable · formalizederdos:488

True ↔ ∀ (A : Finset ℕ), A.Nonempty → 0 ∉ A → 1 ∉ A → ∀ (n m : ℕ), m > n → A.max ≤ ↑n → ↑{x ∈ Finset.Icc 1 m | x ∈ {n | n ≥ 1 ∧ ∃ a ∈ A, a ∣ n}}.card / ↑m < 2 * ↑{x ∈ Finset.Icc 1 n | x ∈ {n | n ≥ 1 ∧ ∃ a ∈ A, a ∣ n}}.card / ↑n

number theoryN/A3 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 proved (Lean) · formalizederdos:493

True ↔ ∃ k N, ∀ (n : ℤ), N ≤ n → ∃ a, (∀ (i : Fin k), 2 ≤ a i) ∧ ∏ i, a i - ∑ i, a i = n

number theoryN/A5 sources

declared proved · formalizederdos:494

∀ (k card p : ℕ), Nat.Prime p → k ∈ Set.Ioo 0 p → p ∣ card → Erdos494.Erdos494Unique k card

analysisadditive combinatoricsN/A2 sources

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