From the elusive nature of prime numbers to the shape of space itself, mathematics reveals itself as a breathtakingly vast and intricately connected universe where simple rules govern profound truths.
Key Takeaways
- Euclid's theorem states that there are infinitely many prime numbers, and the prime number theorem approximates the number of primes less than n as about n / ln(n), with the current record for the largest known prime being 2^136279841 - 1 discovered in 2023.
- Goldbach's conjecture, proposed in 1742, states that every even integer greater than 2 can be expressed as the sum of two primes, verified up to 4 × 10^18 as of 2014.
- The Riemann Hypothesis posits that all non-trivial zeros of the Riemann zeta function have real part 1/2, unsolved since 1859 and central to prime distribution.
- The binomial theorem (x+y)^n = ∑_{k=0}^n C(n,k) x^{n-k} y^k, with C(n,k)=n!/(k!(n-k)!).
- The fundamental theorem of algebra states every non-constant polynomial has a complex root, proved by Gauss 1799.
- A field with p^n elements is unique up to isomorphism, the finite field GF(p^n).
- The circumference of a circle is 2πr, area πr^2, proved by Archimedes.
- Euclid's parallel postulate: through point not on line, exactly one parallel.
- The Pythagorean theorem: in right triangle, a^2 + b^2 = c^2, over 300 proofs.
- The mean value theorem: f continuous [a,b], diff (a,b), exists c with f'(c)=(f(b)-f(a))/(b-a).
- Taylor's theorem: f(x) = ∑_{k=0}^n f^{(k)}(a)/k! (x-a)^k + R_n, Lagrange remainder f^{(n+1)}(ξ)(x-a)^{n+1}/(n+1)!.
- The fundamental theorem of calculus: ∫_a^b f = F(b)-F(a) where F'=f.
- The central limit theorem: sum S_n /√n → N(0,σ^2) standardized.
- Law of large numbers: sample average → expected value almost surely.
- Bayes' theorem: P(A|B)= P(B|A)P(A)/P(B).
The blog explores prime numbers, unsolved conjectures, and key formulas in mathematics.
Algebra
- The binomial theorem (x+y)^n = ∑_{k=0}^n C(n,k) x^{n-k} y^k, with C(n,k)=n!/(k!(n-k)!).
- The fundamental theorem of algebra states every non-constant polynomial has a complex root, proved by Gauss 1799.
- A field with p^n elements is unique up to isomorphism, the finite field GF(p^n).
- The Cayley-Hamilton theorem says every square matrix satisfies its characteristic equation.
- Jordan canonical form decomposes matrices over algebraically closed fields into Jordan blocks.
- The quadratic formula solves ax^2 + bx + c =0: x = [-b ± √(b^2 -4ac)] / (2a).
- Bezout's identity: gcd(a,b)=1 iff exist x,y with ax+by=1, extended Euclidean algorithm finds them.
- The ring of integers Z is a PID, principal ideal domain.
- Galois theory links field extensions to groups; solvable quintics have solvable Galois group.
- The determinant of an n×n matrix is sum over permutations (-1)^sgn(σ) ∏ a_{i,σ(i)}.
- Vector spaces over R have bases; dimension is basis size, invariant.
- The Chinese Remainder Theorem: if moduli coprime, system x≡a_i mod m_i has unique solution mod ∏ m_i.
- Eigenvalues λ satisfy det(A - λI)=0, characteristic polynomial.
- Hilbert's Nullstellensatz: radical of ideal I(V(S)) = I(S) in k[x1..xn], k alg closed.
- The spectral theorem diagonalizes normal matrices over C.
- Polynomial rings k[x1..xn] are UFDs, unique factorization domains.
- The adjugate matrix adj(A) satisfies A adj(A) = det(A) I.
- Symmetric groups S_n have order n!, generated by transpositions.
- Noetherian rings satisfy ascending chain condition on ideals.
- The trace of a matrix is sum of diagonals, invariant under similarity.
- Artin-Wedderburn theorem decomposes semisimple algebras into matrix rings over division rings.
- The rank of a matrix is dimension of column space, ≤ min(m,n).
- Quaternion algebra H over R has basis 1,i,j,k with i^2=j^2=k^2=ijk=-1.
- The kernel of linear map T: V→W is {v | T(v)=0}, subspace.
- Gröbner bases compute ideal membership in polynomial rings.
- The alternating group A_n is simple for n≥5.
- Singular value decomposition A = U Σ V^*, Σ diagonal non-negative.
- The exterior algebra ∧V has dimension 2^dim(V), antisymmetric tensors.
- Cramer's rule: x_i = det(A_i)/det(A) for invertible A.
- Tensor product V⊗W has basis e_i ⊗ f_j.
- The characteristic of a ring is smallest p with p·1=0 or 0.
- PSL(2,7) has order 168, isomorphic to GL(3,2).
- The circle group T = R/Z is divisible abelian.
- Morita equivalence preserves module categories.
Algebra Interpretation
From the binomial theorem's orderly expansion to Galois theory’s elegant symmetry, this list reveals mathematics not as a cold collection of truths, but as a vast, interconnected landscape where numbers dance, shapes conspire, and every abstract structure secretly winks at another across the intellectual horizon.
Analysis
- The mean value theorem: f continuous [a,b], diff (a,b), exists c with f'(c)=(f(b)-f(a))/(b-a).
- Taylor's theorem: f(x) = ∑_{k=0}^n f^{(k)}(a)/k! (x-a)^k + R_n, Lagrange remainder f^{(n+1)}(ξ)(x-a)^{n+1}/(n+1)!.
- The fundamental theorem of calculus: ∫_a^b f = F(b)-F(a) where F'=f.
- e = lim (1+1/n)^n ≈2.71828, ∑ 1/n! from 0 to ∞.
- Riemann integral defined via Darboux upper/lower sums; improper for unbounded.
- The Weierstrass approximation theorem: continuous f on [a,b] uniform limit of polynomials.
- Fourier series ∑ a_n cos(nx) + b_n sin(nx), converges under Dirichlet conditions.
- The gamma function Γ(z) = ∫_0^∞ t^{z-1} e^{-t} dt, Γ(n+1)=n!.
- L'Hôpital's rule: lim f/g = lim f'/g' if 0/0 or ∞/∞ form.
- The Basel problem: ∑_{n=1}^∞ 1/n^2 = π^2 /6, solved by Euler 1734.
- Lebesgue measure on R^n, outer m^*(E)=inf ∑ vol(R_i) covering.
- The monotone convergence theorem: increasing f_n →f pointwise implies ∫ f_n →∫f.
- Green's theorem: ∫_C P dx + Q dy = ∬ (∂Q/∂x - ∂P/∂y) dA.
- The dominated convergence theorem requires |f_n|≤g integrable.
- Stirling's approximation: n! ≈ √(2πn) (n/e)^n.
- Parseval's theorem: (1/π) ∫ |f|^2 = (a_0^2)/2 + ∑ (a_n^2 + b_n^2).
- The heat equation u_t = k u_xx solved by separation of variables.
- Bolzano-Weierstrass: bounded sequence has convergent subsequence.
- The Laplace transform L{f}(s)=∫_0^∞ f(t) e^{-st} dt.
- Fubini's theorem: ∬ f = ∫ dy ∫ f dx under integrability.
- The Cantor set has measure zero, uncountable, dimension log2/log3≈0.6309.
- Stone-Weierstrass: subalgebra separating points dense in C(K).
- The Riemann zeta ζ(s)=∑ 1/n^s for Re(s)>1, analytic continuation.
- Arzelà-Ascoli: equicontinuous bounded pointwise relatively compact.
- The wave equation u_tt = c^2 u_xx, d'Alembert solution.
- Riesz representation: continuous linear functionals on C[0,1] are integrals vs measures.
Analysis Interpretation
From the gentle slope guaranteed by the mean value theorem to the uncountable intricacies of the Cantor set, mathematics is a majestic edifice where continuity bridges the finite to the infinite, approximation polishes the rough, and convergence tames the wild, all held together by theorems that are the quiet, witty punchlines to existence's most persistent questions.
Applied Math
- The central limit theorem: sum S_n /√n → N(0,σ^2) standardized.
- Law of large numbers: sample average → expected value almost surely.
- Bayes' theorem: P(A|B)= P(B|A)P(A)/P(B).
- Binomial distribution B(n,p): P(k)=C(n,k) p^k (1-p)^{n-k}, mean np.
- Poisson distribution λ: P(k)= e^{-λ} λ^k / k!, approximates binomial large n small p.
- Normal distribution N(μ,σ^2): density (1/√(2πσ^2)) exp(-(x-μ)^2/(2σ^2)).
- Chi-squared test statistic ∑ (O_i - E_i)^2 / E_i ~ χ^2_{k-1}.
- Markov's inequality: P(X≥a) ≤ E[X]/a for non-neg X, a>0.
- The birthday problem: probability 2 share birthday in group of 23 is ≈0.507.
- Monte Carlo method estimates π by ratio of points in quarter circle.
- Linear regression: y = β0 + β1 x + ε, β1 = cov(x,y)/var(x).
- Chebyshev's inequality: P(|X-μ|≥kσ) ≤ 1/k^2.
- Entropy H(X)= -∑ p_i log p_i, measures uncertainty.
- The traveling salesman problem is NP-hard, 2^n / n approx solutions.
- PageRank models web links as Markov chain stationary distribution.
- Benford's law: leading digits log10(1+1/d), explains financial data.
- The coupon collector problem: expected trials n H_n ≈ n ln n to collect all n.
- Queuing theory M/M/1: average wait λ/(μ(μ-λ)), ρ=λ/μ<1.
- Simpson's paradox: trend reverses when groups combined.
- The Monty Hall problem: switching wins 2/3 probability.
- p-value is P(data | H0), not P(H0|data).
- Power of test =1 - β, β type II error probability.
- Variance of uniform [a,b]: (b-a)^2 /12.
- Exponential distribution λ: P(X>t)=e^{-λ t}, memoryless.
- Student's t-test for mean, t=(x̄-μ)/(s/√n) ~ t_{n-1}.
- ANOVA F-statistic MS_between / MS_within ~ F_{df1,df2}.
- The hat check problem: derangements !n ≈ n!/e.
- Geometric distribution: trials until first success, mean 1/p.
- Confidence interval for mean: x̄ ± t_{α/2} s/√n.
- Correlation ρ = cov(X,Y)/(σ_X σ_Y), |ρ|≤1.
- The secretary problem: optimal stop at 1/e ≈37%.
- Lyapunov stability: ε>0 δ>0 |t|≤T implies ||φ(t,x)-φ(t,x0)||<ε for ||x-x0||<δ.
Applied Math Interpretation
If you ever wonder why your wildest guesses about reality are eventually crushed into a bell curve, your data’s average settles on truth with stubborn certainty, and why even an all-knowing Bayesian must humbly update their beliefs from evidence, it’s because probability is a trickster god who loves predictable chaos, impossible birthdays, and showing you why you should always switch doors.
Geometry
- The circumference of a circle is 2πr, area πr^2, proved by Archimedes.
- Euclid's parallel postulate: through point not on line, exactly one parallel.
- The Pythagorean theorem: in right triangle, a^2 + b^2 = c^2, over 300 proofs.
- Volume of sphere (4/3)πr^3, surface 4πr^2, by Archimedes' method of exhaustion.
- Euler's formula V - E + F = 2 for convex polyhedra.
- There are 5 Platonic solids: tetrahedron (4 faces), cube (6), octahedron (8), dodecahedron (12), icosahedron (20).
- The Gauss-Bonnet theorem: ∫ K dA + ∫ k_g ds = 2π χ(M) for surfaces.
- Hilbert's third problem: Dehn invariant shows not all polyhedra equidissectable to tetrahedra.
- The seven bridges of Königsberg problem, solved by Euler, origin of graph theory.
- The Banach-Tarski paradox: sphere decomposable into 5 pieces, reassemble into 2 spheres, using axiom of choice.
- The area of equilateral triangle side a is (√3/4) a^2.
- Non-Euclidean geometries: hyperbolic (sum <180°), elliptic (>180°).
- The Mandelbrot set is {c ∈ C : z_{n+1}=z_n^2 + c, z_0=0 stays bounded}.
- Pick's theorem: area = i + b/2 -1 for lattice polygon, i interior, b boundary points.
- The four color theorem: planar maps 4-colorable, proved 1976 by Appel-Haken.
- Fractal dimension of Koch snowflake is log(4)/log(3) ≈1.2619.
- The hairy ball theorem: no continuous tangent vector field on S^2.
- Borsuk-Ulam theorem: continuous f:S^n → R^n has f(x)=f(-x) for some x.
- The Euler line passes through centroid, orthocenter, circumcenter in triangles.
- Viviani's theorem: sum distances to sides equals altitude in equilateral triangle.
- The 3D Szilassi polyhedron has 7 hexagonal faces, each pair adjacent.
- Poncelet's porism: if one n-gon inscribed in conic, tangent to another, all are.
- The regular heptagon not constructible by compass/straightedge, as 2cos(2π/7) not quadratic.
- The Gauss map on convex surfaces has degree related to Euler characteristic.
- The Möbius strip has one side, one boundary, non-orientable.
- The projective plane RP^2 cannot embed in R^3 without self-intersection.
Geometry Interpretation
We have spent millennia tinkering with shapes on pages and in our minds, and it turns out that whether we are wrapping ropes around circles, coloring maps, or cutting spheres into paradoxical pieces, our playful geometry keeps revealing itself to be a universe of profound truths.
Number Theory
- Euclid's theorem states that there are infinitely many prime numbers, and the prime number theorem approximates the number of primes less than n as about n / ln(n), with the current record for the largest known prime being 2^136279841 - 1 discovered in 2023.
- Goldbach's conjecture, proposed in 1742, states that every even integer greater than 2 can be expressed as the sum of two primes, verified up to 4 × 10^18 as of 2014.
- The Riemann Hypothesis posits that all non-trivial zeros of the Riemann zeta function have real part 1/2, unsolved since 1859 and central to prime distribution.
- Fermat's Last Theorem, proved by Andrew Wiles in 1994, states no positive integers a, b, c satisfy a^n + b^n = c^n for n > 2.
- The twin prime conjecture suggests infinitely many primes p where p+2 is also prime, with the largest known pair being (2996863034895 × 2^1290000 ± 1) found in 2016.
- Euler's totient function φ(n) counts integers up to n coprime to n, and ∑_{d|n} φ(d) = n for all n, known since 1763.
- The perfect numbers are rare; the first four are 6, 28, 496, 8128, all even, and Euler proved all even perfect numbers are of form 2^{p-1}(2^p - 1) for prime p.
- Wilson's Theorem states (p-1)! ≡ -1 mod p if and only if p is prime, useful for primality testing.
- The number of partitions of n, p(n), grows asymptotically as exp(π √(2n/3)) / (4n√3) by Hardy-Ramanujan formula.
- There are 10 regular polyhedra (Platonic solids) in 3D, but only 5 convex ones: tetrahedron, cube, octahedron, dodecahedron, icosahedron.
- The Collatz conjecture claims the 3n+1 process reaches 1 for any positive integer, verified up to 2^68 ≈ 10^20.
- Pythagorean triples are solutions to a^2 + b^2 = c^2; primitive ones generated by m>n>0, m-n odd, gcd=1: a=m^2-n^2, b=2mn, c=m^2+n^2.
- The Fibonacci sequence is defined F(0)=0, F(1)=1, F(n)=F(n-1)+F(n-2); F(n) ≈ φ^n / √5 where φ=(1+√5)/2.
- Catalan's conjecture, proved 2002 by Mihăilescu, states 8 and 9 are the only consecutive powers: 2^3 and 3^2.
- The divisor function σ(n) sums divisors of n; abundant numbers have σ(n) - n > n, first is 12 with σ(12)=28.
- Mertens' theorems describe products over primes: ∏_{p≤x} (1 - 1/p) ~ e^{-γ} / ln(x) where γ≈0.57721 is Euler-Mascheroni constant.
- The Pell equation x^2 - d y^2 = 1 has infinitely many solutions for non-square d; fundamental solution generates all via (x1 + y1√d)^k.
- Lagrange's four-square theorem: every natural number is sum of four integer squares, proved 1770.
- The Erdős–Ulam problem asks density of integers with distinct prime factors; unsolved.
- The Ramanujan tau function τ(n) from modular forms; τ(1)=1, τ(2)=-24, and |τ(p)| < 2 p^{11/2} by Deligne's proof.
- There are 2^10 = 1024 subsets of a 10-element set, but only 1 empty and 1 full.
- The harmonic series H_n = ∑_{k=1}^n 1/k ≈ ln(n) + γ, diverges logarithmically.
- Fermat primes are 2^{2^n} +1; known ones: 3,5,17,257,65537; n=5 to 32 composite.
- The Sierpinski number 78557 makes 78557 * 2^n +1 composite for all n≥1.
- The abc conjecture bounds rad(abc) > c^{1+ε} for integers a+b=c coprime, proposed by Masser-Oesterlé.
- Euler's prime generating polynomial n^2 + n + 41 produces primes for n=0 to 39.
- The Landau's problems include Goldbach, twin primes, Legendre's (prime between n^2 and (n+1)^2), unsolved.
- The class number of Q(√-d) is 1 for only 9 negative d: 1,2,3,7,11,19,43,67,163.
- The monster group has order 808017424794512875886459904961710757005754368000000000 ≈ 8×10^53.
- The fundamental theorem of arithmetic states unique prime factorization, proved by Gauss.
Number Theory Interpretation
The universe is built on the elegant, unbreakable rules of primes—like Euclid’s infinite list and Goldbach’s stubborn even sums—but it also delights in leaving us teasing mysteries, such as the Riemann Hypothesis’s hidden zeros and the Collatz conjecture’s deceptively simple loop.