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Some theorems

@CihanPostsThms

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Posting some theorems, and occasionally other stuff. By @bahran_cihan

Joined July 2022

Don't wanna be here? Send us removal request.

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@CihanPostsThms

Some theorems

@CihanPostsThms

2 years

NON-THEOREM ANNOUNCEMENT: As someone struggling with depression/isolation, one reason I opened this account is to get some sense of belonging with the math community. I would appreciate if you can refrain from "that is obvious duh", "you miss the point" type of comments. Thanks.

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1 year

[Cantor 1873]: The quadratic function P: ℕ x ℕ → ℕ (x,y) ↦ ½( (x+y)² + 3x + y ) is a bijection. [Fueter–Pólya 1923]: The only quadratic bijections ℕ x ℕ → ℕ are Cantor's P(x,y) and P(y,x).

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[Matiyasevich 2020]: The below Python program halts if and only if the Riemann hypothesis is false:

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Recent finding in Conway's game of life: it contains a finite ancient configuration. More precisely, if the following 30 x 24 configuration occurs anywhere in a given universe, it must have been there since the beginning of time.

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A subset A ⊆ ℝ² is called a *cloud* around 𝑥 if every line through 𝑥 has a finite intersection with A. [Komjáth 2001]: The following are equivalent: (1) Three clouds cover ℝ². (2) The continuum hypothesis holds.

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0⁰ = 0! = 1 . Proof: There is exactly one map from the empty set to itself and it is a bijection.

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For every n, consider the n x n matrix R(n) whose (i,j) entry is 1 if j=1 or i divides j, and 0 otherwise. [Redheffer 1977]: The following are equivalent: (1) For every ε>0 we have |det R(n)| = O(n^(1/2 + ε)). (2) The Riemann hypothesis holds.

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Please include more words than "where", "for", "and", "odd" in your theorem statements.

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Borwein integrals:

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[Morley 1899]: In any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle.

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7 months

196884 = 196883 + 1 .

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[Volchkov 1995]: Writing 𝛾 for the Euler–Mascheroni constant, TFAE: (1) ∫(1-12t²)/(1+4t²)³·∫log |𝜁(𝜎+it)| d𝜎 dt = 𝜋(3-𝛾)/32 where the first integral is from 0 to ∞ and the second is from 1/2 to ∞. (2) The Riemann hypothesis holds.

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Let V be a vector space over a field F with |V| < ∞. The following are equivalent: (1) The sum of all vectors in V is nonzero. (2) |V| = 2.

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Mathematics. "Every person in your company is a vector. Your progress is determined by the sum of all vectors." ~Elon Musk,

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The fundamental theorem of calculus: The derivative defines a short exact sequence 0 → ℝ → C∞(ℝ,ℝ) → C∞(ℝ,ℝ) → 0 which splits on the right via the definite integral.

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2 years

The infinite product Π (p²-1)/(p²+1) , as p ranges over the prime numbers, is equal to 2/5.

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[Elsholtz 2020]: There exists a constant A = 1.00536773279814724017... such that for every n ∈ ℕ⁺, the number ⌊ A^(10¹⁰ⁿ) ⌋ is prime.

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There is no algorithm that takes two 15x15 integer matrices A and B as input and decides whether a (finite) product of A's and B's equals the zero matrix.

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There is no continuous function f: ℂ² → ℂ such that ∀b,c ∈ ℂ, f(b,c) is a root of the quadratic equation x² + bx + c = 0.

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[Churchill–Biderman–Herrick 2020]. 𝐌𝐚𝐠𝐢𝐜: 𝐓𝐡𝐞 𝐆𝐚𝐭𝐡𝐞𝐫𝐢𝐧𝐠 is Turing complete.

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𝐏𝐢𝐜𝐤'𝐬 𝐓𝐡𝐞𝐨𝐫𝐞𝐦: Let P be a polygon in ℝ² whose vertices lie in the lattice ℤ². Writing Int(P) for the interior of P and ∂P for the boundary of P, we have Area(P) = |Int(P) ⋂ ℤ²| + ½|∂P ⋂ ℤ²| - 1 .

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The volume of the unit d-sphere goes to 0 as d → ∞. It is maximized at d=5.

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Theorem 2.1 of Gauld's book "Non-metrisable manifolds" lists 100+ equivalent conditions for a connected Hausdorff locally Euclidean space to be metrizable:

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∫_ℝ [cos(x)/(1+x²)] dx = 𝜋/𝑒 .

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The smallest solution of the equation a/(b+c) + b/(a+c) + c/(a+b) = 4 in positive integers have each a,b,c around 80 digits.

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For 𝑎 ∈ ℝ⁺, the following are equivalent: (1) For every x ∈ ℝ, we have 𝑎^x ≥ x+1. (2) 𝑎 = 𝑒 .

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[Johann Bernoulli 1697]: The integral ∫x^(-x)dx as x goes from 0 to 1, is equal to the sum of the series Σ n⁻ⁿ as n goes from 1 to ∞.

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The density of 2x2 integer matrices diagonalizable over ℝ inside all 2x2 integer matrices is 49/72.

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[Gualà–Leucci–Natale 2014]. Candy Crush on an (m x n) board is NP-Hard.

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[Rademacher 1937]: The function p(n) := the number of partitions of n satisfies the following formula:

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[Erdős 40s?]: The following are equivalent: (1) Every coloring of the real numbers by countably many colors has a monochromatic solution to the equation x₁ + x₂ = x₃ + x₄ with distinct x_i. (2) The continuum hypothesis is false.

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Among the standard axioms for a unital ring, the commutativity of addition is superfluous. Proof: x + (y+x) + y = (x+y) + (x+y) = (1+1)(x+y) = (1+1)x + (1+1)y = (x+x) + (y+y) = x + (x+y) + y .

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[Stallings 1962]: A sufficiently associative infinite product on a group is trivial.

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A Diophantine characterization of the factorial that goes through the alphabet:

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Every function from ℝ to ℝ is a sum of two injections.

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[Euler 1744]. 𝑒 is irrational. [Hermite 1873]. 𝑒 is transcendental.

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Consider a d-cube with side length 4. Inscribe 2ᵈ many unit d-spheres in it as depicted. Next, put a middle sphere tangent to each of these unit spheres. The strange thing is, the middle sphere has radius √d - 1, and hence when d ≥ 10 manages to protrude outside the cube!

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[Agronsky–Bruckner–Laczkovich–Preiss 1985]: For every continuous f: [0,1] → ℝ, there exists a continuously differentiable g: [0,1] → ℝ such that the set {x ∈ [0,1] : f(x) = g(x)} is uncountable.

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[Bagemihl 1951]: Let ℒ be the set of straight lines in ℝ². Given any function f: ℒ → {2, 3, 4, ...} , there is a subset X ⊆ ℝ² such that for every line L ∈ ℒ we have | X ⋂ L | = f(L) .

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[Hasse 1966]: Given x>0, let P(x) be the set of prime numbers ≤ x and A(x) := {p∈P(x) : p divides 2ⁿ+1 for some n∈ℕ}. Then lim_{x→∞} |A(x)| / |P(x)| = 17/24.

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Given sets A,B, write B^A for the set of functions A→B. Given cardinal numbers λ,κ, choosing sets A,B with |A| = λ, |B| = κ, define κ^λ := |B^A|. Theorem. 0⁰ = 1 .

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Writing 𝜙 for the golden ratio, this integral equals 4𝜋⋅arccot(√𝜙) .

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[Hölder 1887]: There is no nonzero polynomial differential equation (in the variables x, y, y', y'', ...) that the Gamma function y = Γ(x) satisfies.

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Given a > 0, the power tower a^a^a^a... , that is, the limit of the sequence A[n] given by A[1] := a, A[n] := a^A[n-1], exists if and only if e^(-e) ≤ a ≤ e^(1/e) .

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It is possible to divide by 3 without the axiom of choice. More precisely, it is a theorem of ZF that 3 x A ≅ 3 x B implies A ≅ B for any sets A,B.

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1 year

There does not exist an algorithm which takes a positive integer K as input and decides whether the following (Diophantine) equation has a solution over non-negative integers:

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[Komjáth 1994]: The points in ℝⁿ can be colored by countably many colors in such a way that no two points at a rational distance get the same color.

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Up to oriented diffeomorphism, - There is a unique smooth structure on the 61-sphere. - There are exactly 24 smooth structures on the 62-sphere. - There are exactly 142211872163171481167115958878208 smooth structures on the 63-sphere.

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[Booker–Sutherland 2019]: 42 is a sum of three cubes. More precisely, 42 = (-80538738812075974)³ + (80435758145817515)³ + (12602123297335631)³ .

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1 year

For n∈ℕ, 𝜎(n) := the sum of the divisors of n. 𝛾 := Euler–Mascheroni constant A := { 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 36, 48, 60, 72, 84, 120, 180, 240, 360, 720, 840, 2520, 5040 } B := { n≥3: 𝜎(n) / [n∙log(log(n))] ≥ exp(𝛾) } We know that A ⊆ B.

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10 months

I feel so powerless and I am ashamed of some of the stuff I see in my timeline. If you think a 24 hour notice to "evacuate" (no guarantee of return) a region of 1.1 million (including disabled ppl etc.) over broken roads is fair, please unfollow me. This is genocide theatrics.

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A subset X ⊆ ℝ² is called linearly accessible if for every x ∈ X there exists a straight line L ⊆ ℝ² such that L ∩ X = {x} . [Nikodym 1927]: There exists a linearly accessible Borel set X ⊆ [0,1]² such that the complement [0,1]² – X has Lebesgue measure zero.

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2 years

The following statement is independent of ZFC: "Every pair of sets X,Y with |X| < |Y| satisfies |2^X| < |2^Y|."

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[Nevanlinna 1929]: Let Ψ ⊆ ℂ ∪ {∞} with |Ψ| ≥ 5. If f,g : ℂ → ℂ ∪ {∞} are non-constant meromorphic functions with inverse image sets satisfying f⁻¹(c) = g⁻¹(c) for every c∈Ψ, then f = g. ----- Here the lower bound 5 cannot be improved.

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2 years

The nine-point circle: For any triangle, the midpoint of each side, the foot of each altitude, and the midpoint between every vertex and the orthocenter lie on a single circle.

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An m ✕ n chessboard with m ≤ n has a knight's tour if and only if it satisfies the following three conditions: (1) At least one of m,n is even. (2) m ∉ {1, 2, 4}. (3) (m,n) ∉ {(3,4), (3,6), (3,8)}.

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In constructive mathematics, the following are equivalent: (1) The law of excluded middle holds. (2) Every subset of {0} is finite.

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[Kunis–Reichenwallner–Reitzner 2018]: There are convex sets L ⊆ K ⊆ ℝ³ such that the expected volume of the convex hull of 4 points chosen uniformly random in L is strictly greater than that in K.

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1 year

[Robin 1984]: TFAE: (1) B – A ≠ ∅. (2) B is infinite. (3) The Riemann hypothesis is false.

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2 years

Counted up to isomorphism, more than 99% of groups of order ≤2000 have order 1024.

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1 year

Relevant lore:

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[Ball–Rivoal 2001]. For every odd integer m≥5, the dimension of the ℚ-vector space generated by 1, 𝜁(3), 𝜁(5), ..., 𝜁(m) ∈ ℝ is at least ⅓⋅log(m).

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11 months

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@shortstein

Thomas Steinke

11 months

This quote sums up why I'm hesitant to make any claims/predictions about what AI cannot do.

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Write R := {m ∈ ℕ odd : m ≠ a²+b²+10c² for any a,b,c ∈ ℕ}. R' := {3, 7, 21, 31, 33, 43, 67, 79, 87, 133, 217, 219, 223, 253, 307, 391}. [Ramanujan 1917]. R' ⊆ R. [Duke–Schulze-Pillot 1990]. R is finite. [Ono–Soundararajan 1997]. GRH implies R = R' ⋃ {679, 2719}.

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That is not mind-blowing.

@elonmusk

Elon Musk

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@waitbutwhy Mind-blowing concept when you first hear it. There is arguably a practical maximum to digits of pi, which is the number required to divide the universe into cubes of Planck length.

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2 years

Writing F(n) for the nth Fibonacci number (F(1) = F(2) = 1), we have Σ 10⁻⁽ⁿ⁺¹⁾⋅F(n) = 1/89 .

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[Graham–Rothschild–Straus 1974]. Given n∈ℕ, TFAE: (1) There exists S ⊆ ℝⁿ with |S| = n+2 so that for any distinct x,y ∈ S, the Euclidean distance d(x,y) is an odd integer. (2) n ≡ 14 (mod 16).

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Let κ be a cardinal number. Choosing any set X of cardinality |X| = κ, define κ! := |Sym(X)|. Theorem: 0! = 1 .

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4 months

[Breiding–Sturmfels–Timme 2020]. There are exactly 3264 conics in ℝ² tangent to the 5 conics given by the equations A(x,y) = 0 B(x,y) = 0 C(x,y) = 0 D(x,y) = 0 E(x,y) = 0 as follows:

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Fix a parameter q. For n,k∈ℕ, define [n] := 1 + q + ⋯ + qⁿ⁻¹ , [n]! := [1]·[2]· ⋯ ·[n] , [n choose k] := [n]! / ([k]! · [n-k]!) , all in ℤ[q]. Theorem. Let V be an n-dimensional vector space over a finite field F_q. Then [n choose k] = | {W ≤ V : dim(W) = k} | .

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The 3x3x3 Rubik's cube group is 2-generated.

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2 years

Very illuminating final step of the proof:

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[Chen 1973]. There are infinitely many primes p such that p+2 is either prime or a product of two primes.

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Every countable group is the fundamental group of an open connected subset of ℝ⁴.

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[Mihăilescu 2002]: The only consecutive pair of proper powers of positive integers is (2³,3²)=(8,9).

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Theorem. For n≥3, the n-th root of 2 is irrational. Unfortunately circular "proof". For a,b∈ℕ, the equation (a/b)ⁿ=2 gives bⁿ + bⁿ = aⁿ, contradicting Fermat's last theorem.

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[Minkowski 1889]: Let C �� ℝⁿ be convex and symmetric with respect to the origin 0⃑. If vol(C) > 2ⁿ, then C ⋂ (ℤⁿ- {0⃑}) ≠ ∅ .

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Every natural number is a sum of at most 9 cubes.

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Theorem: The full permutation group of the natural numbers Sym(ℕ) has cardinality ≥ |ℝ|. Proof: Fix a conditionally convergent series Σ a_n such as a_n = (-1)ⁿ/n. Now Riemann rearrangement thm defines a surjection Sym(ℕ) → ℝ⊔{✦} σ ↦ Σ a_σn if convergent ↦ ✦ o.w.

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[Rosen 1961]. There is a space X such that X x ℝ is homeomorphic to ℝ⁴, but X has no open subset homeomorphic to ℝ³.

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[Bing 1959]: There exists a space X which is not even a manifold, yet X x ℝ is homeomorphic to ℝ⁴.

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It is a theorem of ZF that |ℝ| ≤ |ℝ/ℚ|. Moreover, ZF has models in which |ℝ| < |ℝ/ℚ|.

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[Siebeck 1864, Marden 1945]: Let p(z) ∈ ℂ[z] be a cubic polynomial whose roots z₁, z₂, z₃ are non-collinear points in ℂ, forming a triangle T. Then the foci of the unique ellipse inscribed in T and tangent to the sides of T at their midpoints are the roots of p'(z).

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[Pazzis 2021]: Every automorphism of an infinite-dimensional vector space over a field is the product of 4 involutions.

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As an element of the power series ring ℤ[[t]], the polynomial t² + 3t + 2 = (t+1)(t+2) is irreducible, while t+6 is reducible.

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2 years

A universal property for [0,1] that (arguably) justifies its use in the definition of homotopy:

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Some theorems

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2 years

[Maynard 2019]: For every d ∈ {0,1,2,3,4,5,6,7,8,9}, there are infinitely many primes which do not have d in their decimal expansion.

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7 days

If f: ℂ→ℂ is entire and f is not a polynomial, then the set {z∈ℂ : f⁻¹{z} is a finite subset of ℂ} has at most one element.

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10 months

Given an element g of a group G, write |g| ∈ ℤ⁺ ⋃ {∞} for its order, and set ω(G) := {|g| : g∈G}. [Lytkina–Kuznetsov 2007]. Given a group G, TFAE: (1) ω(G) = {1, 2, 3, 4, 7}. (2) G ≅ PSL₂(7).

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2 years

A form of "continuous induction": Let a < b ∈ ℝ. Say a subset S ⊆ [a,b] is 𝐢𝐧𝐝𝐮𝐜𝐭𝐢𝐯𝐞 if: (1) a ∈ S. (2) For x ∈ [a,b), x ∈ S ⇒ [x,y] ⊆ S for some y > x. (3) For x ∈ (a,b], [a,x) ⊆ S ⇒ x ∈ S. Theorem: A subset S ⊆ [a,b] is inductive ⇔ S = [a,b].

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Some theorems

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2 years

[Kassabov 2010]: For every n ≥ 2, the ring of (n x n) matrices over ℤ has a presentation with 2 generators and 3 relations: ⟨x,y : xⁿ = yⁿ = 0, xy + yⁿ⁻¹xⁿ⁻¹ = 1⟩

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Some theorems

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2 years

[Mohr 1672]: Every (pointwise) geometric construction that can be performed with a compass and straightedge can be done with a compass alone.

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2 years

[Tseitin 1958]: For the semigroup presentation ⟨𝑎,𝑏,𝑐,𝑑,𝑒 : 𝑎𝑐=𝑐𝑎, 𝑎𝑑=𝑑𝑎, 𝑏𝑐=𝑐𝑏, 𝑏𝑑=𝑑𝑏, 𝑒𝑐𝑎=𝑐𝑒, 𝑒𝑑𝑏=𝑑𝑒, 𝑐𝑑𝑐𝑎=𝑐𝑑𝑐𝑎𝑒, 𝑐𝑎𝑎𝑎=𝑎𝑎𝑎, 𝑑𝑎𝑎𝑎=𝑎𝑎𝑎⟩ there is no algorithm to decide whether a given word is equivalent to 𝑎𝑎𝑎 or not.

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Some theorems

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2 years

[Lagarias 2002]: Let H(n) := 1 + 1/2 + ⋯ + 1/n be the n-th harmonic number. The following are equivalent: (1) For every n ≥ 2, the sum of the (positive) divisors of n is strictly less than H(n) + exp[H(n)]·log[H(n)] . (2) The Riemann Hypothesis is true.

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1 year

Given a set X, write F(X) for the free group on X. [Kleppmann 2014]. Over ZF, TFAE: (1) For every pair of infinite sets X,Y, we have |F(X)| = |F(Y)| ⇒ |X| = |Y|. (2) The axiom of choice holds.

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