Mathematweets

Mathematical tweets.
Σᵢ∈ₓ|Tᵢ|≥n|X|+m⇒∀W⊆X∃S⊆X:(|W|=|S|∧Σᵢ∈ₛ|Tᵢ|≥n|W|+min(m,|W|)).🔣
Σᵢ∈ₓ|Tᵢ|≤n|X|+m⇒∀W⊆X∃S⊆X:(|W|=|S|∧Σᵢ∈ₛ|Tᵢ|≤n|W|+max(0,m+|W|-|X|)).

[Mathematweets @mathematweets]

Conjecture of the day: Fortune's conjecture. For an integer n>0, let p be the n-th primorial (i.e. product of first n primes) and q be the least prime greater than p, then q-p is either 1 or prime. #numbertheory #prime #primorial #math

[Mathematweets @mathematweets]

Theorem of the day: n=(6k+1)(12k+1)(18k+1) is a Carmichael number (i.e. for any integer b, n divides bⁿ-b) if 6k+1, 12k+1, 18k+1 are all prime. Proven by Chernick in 1939. #math #numbertheory #prime #Carmichael

[Mathematweets @mathematweets]

Theorem of the day: A regular polygon of n sides can be constructed by ruler, compass & angle-trisector if and only if n=2ʳ3ˢp₁p₂...pₖ where pᵢ>3 are distinct primes of the form 2ᵗ3ᵘ+1. Proven by Gleason in 1988 & extends the Gauss-Wantzel theorem https://t.co/Cnl8rRGL6f

[Mathematweets @mathematweets]

Gauss-Wantzel: regular n-gon constructible using compass and straightedge iff product of power of 2 & (possibly nil) distinct Fermat primes.

[Mathematweets @mathematweets]

Conjecture of the day: Oppermann's conjecture (1877). For a positive integer n, there exists a prime between n(n+1) and (n+1)² and there exists a prime between (n+1)² and (n+1)(n+2). #math #numbertheory #prime

[Mathematweets @mathematweets]

Theorem of the day: Mills' formula (1947). There exists a real number A such that ⌊A^(3ⁿ)⌋ is prime for all n. If the Riemann Hypothesis is true, then the smallest such A is 1.30637788.... #numbertheory #math #prime

[Mathematweets @mathematweets]

Theorem of the day: Nicomachus' theorem. 1³+2³+...+n³ = (1+2+...+n)². #math #numbertheory

[Mathematweets @mathematweets]

Oskar Perron (of the Perron-Frobenius theorem fame, among others) gave a proof of the correctness of this algorithm in 1951. #math #algebra

[Mathematweets @mathematweets]

Theorem of the day: Moessner's magic (1951). For a fixed n, start with 1,2,3,... , and for i=0...n-2, at step i, remove every (n-i)-th element of the sequence, and compute the partial sums. The result after i=n-2 is the sequence 1ⁿ, 2ⁿ, 3ⁿ, ... #math #algorithm

[Mathematweets @mathematweets]

Theorem of the day: Moessner's magic (1951). For a fixed n, start with 1,2,3,... , and for i=0...n-2, at step i, remove every (n-i)-th element of the sequence, and compute the partial sums. The result after i=n-2 is the sequence 1ⁿ, 2ⁿ, 3ⁿ, ... #math #algorithm

[Mathematweets @mathematweets]

Theorem of the day: Shoelace, surveyor or Gauss' area formula. A simple polygon with a counter clock-wise sequence of vertices (xᵢ,yᵢ), i=1..n has area ((y₁+y₂)(x₁-x₂)+...+(yₙ+y₁)(xₙ-x₁))/2=(x₁y₂-x₂y₁+x₂y₃-x₃y₂+...+xₙy₁-x₁yₙ)/2. #math #geometry #polygon

[Mathematweets @mathematweets]

Theorem of the day (Crocker, 1971): If n>2, then 2²^ⁿ-1 cannot be written as p + 2ᵃ + 2ᵇ where p is prime and a ≠ b. #math #numbertheory

[Mathematweets @mathematweets]

Meant to say Besge's equation. #math

[Mathematweets @mathematweets]

Theorem of the day: Besge's inequality (1862). Σᵢ₌₁ⁿ⁻¹ σ(i)σ(n-i) = [5σ₃(n)+(1-6*n)σ(n)]/12 where σ(n) is the sum of the divisors of n and σ₃(n) is the sum of the cubes of the divisors of n. #numbertheory #math

[Mathematweets @mathematweets]

Theorem of the day: Blichfeldt's theorem (1914). A bounded set of area A in the Euclidean plane can be translated so that at least ⌈A⌉ points of the set have integer coordinates. #math #geometry

[Mathematweets @mathematweets]

Theorem of the day: generalized Pick's thm. For a 2D polygon whose vertices are on a grid with integer coordinates, its area is a+½b+h-1, where a,b are the number of grid points in the interior & boundary resp. & h is the number of holes. https://t.co/pJkJhgs4tm #geometry #math

[Mathematweets @mathematweets]

Theorem of the day: Pick's theorem (1899). For a simple polygon on the 2D grid of points with integer coordinates whose vertices are grid points, its area is A=a+½b-1, where a,b are the number of grid points in the interior & boundary resp. Thus 2A is an integer. #geometry #math

[Mathematweets @mathematweets]

Theorem of the day (Crocker, 1971): If n>2, Bⱼ>1 divides 2²^ʲ+1 for j=0..n-1, and (16k+1)∏ⱼBⱼ ≤ 2²^ⁿ-1, then (16k+1)∏ⱼBⱼ cannot be written as p + 2ᵃ + 2ᵇ where p is prime and a ≠ b. This generalizes https://t.co/c0r7HY7vxz #math #numbertheory

[Mathematweets @mathematweets]

Theorem of the day (Crocker, 1971): If n>2, then 2²^ⁿ-1 cannot be written as p + 2ᵃ + 2ᵇ where p is prime and a ≠ b. #math #numbertheory

[Mathematweets @mathematweets]

Theorem of the day (Crocker, 1971): If n>2, then 2²^ⁿ-1 cannot be written as p + 2ᵃ + 2ᵇ where p is prime and a ≠ b. #math #numbertheory

[Mathematweets @mathematweets]

Theorem of the day: Robin's theorem (1984). The Riemann hypothesis is true if and only if the sum of the divisors of n < eᶜ n log(log(n)) for all n>5040 where c≈0.5772157 is the Euler-Mascheroni constant. #numbertheory #math #Riemann #Euler

[Mathematweets @mathematweets]

Theorem of the day: Dixon's identity (1891). Σᵃₖ₌₋ₐ (-1)ᵏ C(2a,k+a)³ = (3a)!/a!³ where C(n,k) is the binomial coefficient of n choose k.
https://t.co/G0dADiM5tG
#math #combinatorics

[Mathematweets @mathematweets]

Theorem of the day: Bernoulli's inequality. If x+1≥0, then (1+x)ʳ≤1+rx for 0≤r≤1 and (1+x)ʳ≥1+rx for r≥1. https://t.co/pdKCUzkt0z
#math #calculus #analysis

[Mathematweets @mathematweets]

Theorem of the day: Rolle's theorem. Let f be a continuous real function on [a,b], differentiable on the interior (a,b) with f(a)=f(b). Then there exists a<c<b such that f'(c) = 0. This is a consequence of the mean value theorem. https://t.co/i44rlwdn9y #math #calculus #analysis

[Mathematweets @mathematweets]

Theorem of the day: Mean value theorem. Let f be a real-valued function continuous on [a,b] and differentiable on (a,b), then there exists c such that a<c<b and f'(c)(b-a) = f(b)-f(a), i.e. the tangent line at c is parallel to the line through (a,f(a)) & (b,f(b)). #math #calculus