For centuries, the gold standard for mathematical reasoning wasn’t just inspired by Euclid’s Elements – it was Euclid’s Elements. Compiled around 300 BCE, this monumental 13-book collection systematically derived a vast body of geometry and number theory from a small set of explicit starting points. It begins with fundamental plane geometry (Book I covers basic […]
How do we construct extensions with given Galois groups?
Extensions of what? How are we given the Galois groups? How do we choose to view a given group to construct these extensions? Let’s say rationals. And some finite group. Start with a basic group like a cyclic group. How do we construct cyclic extensions? How do they look like? We have to start with […]
Sophie Germain’s Theorem
As early attempts to the proof of Fermat’s last theorem, many mathematicians solved the problem for small exponents. While these special cases are being studied, Sophie Germain, a French mathematician, came up with the following interesting result. (Look at https://www.agnesscott.edu/lriddle/women/germain.htm for her fascinating and revolutionary story) Theorem 1: For any odd prime such that is […]
Gaps between Farey fractions
Consider the rationals (fractions) between and with denominator bounded by . That is These fractions are called Farey fractions of level and we denote them by The number of fractions is Neighbours: Any two neighbours have the property that In fact, two fractions are adjacent to each other in some Farey sequence iff . In […]
Topological Spaces and Continuity
Our first intuition of continuity is usually geometric. A function is continuous at a point if small changes in the input produce small changes in the output. If a point is very close to , then should be very close to . This is the picture we inherit from functions on the real line: the […]
Furstenberg’s Topological Proof of the Infinitude of Primes
One surprising proof of the infinitude of primes is Furstenberg’s topological proof. At first glance it looks like a clever trick: put a strange topology on the integers, observe that arithmetic progressions are open and closed, and then use the fact that a finite union of closed sets is closed. But the proof is more […]
The one dimensional Ising Model
The one-dimensional Ising model consists of a row of spins, each of which can point in one of two directions. We label these two directions by and . The essential feature is that neighboring spins interact: when the coupling is ferromagnetic, neighboring spins prefer to agree. Thus two adjacent positive spins and two adjacent negative […]
Three Squares Theorem by Geometry of Numbers
Three Squares Theorem: if is a positive integer not of the form , then is the sum of three squares. We present a proof using geometry of numbers due to Ankeny. We prove it for the case , and squarefree. Let Find a prime such that for all . Therefore we find solutions to implies […]
Schönemann’s Proof of Irreducibility of Cyclotomic Polynomials.
Problem: Show that the cyclotomic polynomial is irreducible. The standard presentation of irreducibility is by Eisenstein’s criterion: Consider the shift Now observing that every term is and the last term is , we are done by Eisenstein’s criterion. We give an alternate proof by Schonemann. Schonemann’s proof of irreducibility of For a prime consider the […]
Legendre’s three-square theorem
A natural number can be written can sum of three squares, if and only if is not of the form for nonnegative integers and Modulo every square has to be , or and hence cannot be Also if divides then all have to be even. These two facts prove the only if part. Now to […]
Epsilon-biased Sets and Derandomized Linearity Testing
In this post, we will show that choosing one of the vectors from a epsilon-biased set (that is pseudorandom for for statistics like bias of parity functions), the linearity test works. Constructing small epsilon biased sets reduces the size of our sample space for the randomness used in the BLR test. A set is called […]
Erdos-Selberg Elementary PNT
Elementary Prime number Theorem1. Selberg symmetry identity : Proof: 2. Tauberian argument Iterative arguments- Using Brun-Titchmarsh Relation to the analytic proof and non-vanishing of zeta(s) on Where are the zeroes?Smoothing of mobius- to get these higher Von Mangoldt functions.Relation to Chebyshev’s estimates? Approximations to mobius.. Error terms in this elementary proof- use generalized Selberg identities […]
BLR Linearity testing
A function is linear if either of the following conditions hold: for some (Global description)For all (Local description) What can we say about function which satisfies many of these local requirements? How close is it to a linear function? How does the distance to linear functions related to the fraction of local requirements satisfied? BLR […]
Clairut’s Relation: Geodesics on Surfaces of Revolution
One of the recurring themes in mathematics is how symmetry simplifies problems. In differential geometry, surfaces of revolution – shapes like spheres, cylinders, cones, or donuts, formed by spinning a curve around an axis – possess a fundamental rotational symmetry. It turns out this symmetry provides a powerful shortcut for understanding the “straightest paths,” or […]
Erdos-Selberg PNT Historical Survey
Following is a historical survey by Dorian Goldfeld about the elementary proof of prime number theory and the Erdos-Selberg dispute.
Isoperimetric Inequality
Minimum length of the curve bounding an area A? Maximum area bound by a closed curve of length L? Optimal case: Circle. Steiner Proof: Consider a curve with given perimeter 1. If the region bound by the curve is not convex – there is a chord joining two point on the curve which lies outside […]
Groups, Categories, Representations
A way to think of groups is that they correspond to symmetries. Symmetries of an object satisfy some properties which exactly correspond to the group axioms. An equivalence between an object and its other transformed form is what we mean by symmetry. Act of doing nothing to the object does not transform the object. It […]
Peg Solitaire Invariants
The peg solitaire (Hi-Q) is a game defined by moves where one can move a peg orthogonally over an adjacent peg to an empty peg while removing the peg that’s jumped over. One can ask many question about the game, about the possible states that can be reached from a given initial state or about […]
Volumes of Spheres
When we first learn geometry, spheres feel completely intuitive. A circle in the plane. A ball in three-dimensional space. Everything is visual. High-dimensional geometry behaves in ways that feel almost paradoxical. Volumes shrink, surfaces dominate interiors, and many familiar formulas suddenly depend on special functions like the Gamma function. Understanding why requires stepping away from […]
Legendre, Jacobi, Kronecker Symbols
For an odd prime , the Legendre symbol is defined as the quadratic residue symbol, This is a character modulo and is helpful as a “harmonic” in contrast to the Gauss’s notation and which serve as indicators for being a quadratic residue and non-residue. We also distinguish and rest of the quadratic residues. Some properties […]
Class Number Formula
The class number formula is one of the beautiful results in number theory. It connects the arithmetic of a number field with the behavior of an analytic function at . On the arithmetic side stand the class number, the units, the regulator, the discriminant, and the roots of unity. On the analytic side stands the […]
Gamma Function: Duplication, Multiplication, Reflection Formulae
Weierstrass: Factorials (Shift Identity): Proof: Integration by parts! Relection Formula: (Euler) Proof 1: Start with any product expansion fo But we have Note by differentiating twice, this is equivalent to showing and this can be established by noting that the difference of both sides is a bounded analytic function. Therefore Proof 2: Beta function: Proof: […]
The average fractional part of x/p
For a real number , write for its fractional part. Assuming the prime number theorem, we shall prove that where is Euler’s constant. The quantities fluctuate in a seemingly irregular way as runs through the primes. The useful observation is that, after dividing by , the primes up to become uniformly distributed through with respect […]
Brouwer’s fixed-point theorem
We prove that every continuous map from the closed unit ball to itself has a fixed point. The theorem is usually presented in topological language: a fixed-point-free map would produce a retraction of the ball onto its boundary, but the ball cannot retract onto its boundary. The proof below makes that mechanism visible. The key […]
Abel’s theory of equations solvable by radicals, Abelian Transformations
Abel’s proof that the general quintic cannot be solved by radicals is sometimes presented as though it ended the classical theory of algebraic equations. In fact, it posed a new and more precise problem. Once one knows that no universal radical formula can solve an arbitrary equation of degree five, the natural question is no […]
QR algorithm
The QR algorithm is a method for finding eigenvalues. In the real symmetric case, the problem is especially clean. We are given and we want to find numbers and orthonormal vectors such that . Equivalently, we want an orthogonal matrix and a diagonal matrix such that Thus the eigenvalue problem is, at heart, a problem […]
Abel’s proof of insolvability of the quintic
For more than two centuries, the solution of polynomial equations had appeared to follow a compelling pattern. The quadratic equation had a formula. The cubic equation had yielded to the methods of the Italian algebraists, and the quartic had soon followed. Each success had the same general character: starting from the coefficients, one combined the […]
Jacobi Sums and Fermat’s theorem on Sums of Squares.
Look at the Gauss Sum for a multiplicative character It’s easy to see by executing the double sum (with a change of variables) that (determining sign is more harder- you can apply poisson summation or Gauss’s -binomial identities etc to determine the sign) Consider the Jacobi Sums. Gauss Sums are the analogues of the Gamma […]
Sums of Squares – Continued Fractions
Given a prime we want to find solutions to Method 1: Find the solution of where Expand into a simple continued fraction to the point where the denominators of its convergents satisfy the inequality Then Proof: For any , we have which is true for any continued fraction expansion. Let Take We have Therefore Method […]
Quadratic Irrationals, Continued Fractions, Pell’s Equation..
Quadratic irrational are solutions to quadratic equations with integer coefficients. Continued fractions: Expansion of the form If then Relation to GL_2 (invertible integers matrices): Quadratic Irrationals and periodic continued fractions: Theorem: An infinite integral continued fraction is periodic if and only if it represents a quadratic irrational. Proof: (Easy) Multiplying it out, we see satisfies […]
Gershgorin Disks and Bounds on Roots of Polynomials
Gershgorin Disks Let be a complex matrix. Define be the sum of non-diagonal entries of the -th row. be the closed disk of radius around These are called Gershgorin disks. Theorem: Every eigen value of is contained in some Gershgorin disk Proof: Given an eigenvalue , take an eigenvector and normalize to make the largest […]
Gauss: Arithmetic-Geometric Mean, Elliptic Functions, Approximations to Pi
Gauss-Legendre algorithm: The validity of the above algorithm can be seen as a consequence of the following fact: Proof of this formula: It will follows from the relation between Arithmetic-Geometric mean and the elliptic integrals. In fact, it it closely related to Legendre relation between elliptic integrals of first and second kind. Details: Arithmetic-Geometric Mean: […]
Squares in Progressions
Find all arithmetic progressions where are all squares. So we need to find integers such that ie., Hence we get a rational point on the circle is a point on the circle and now using a pencil of lines passing though parametrized by slope , we see that all the point are given by What […]
Multiplication Algorithms
Multiplication How do we multiply numbers? Assume that we the input representation of a number is in decimals. How can we compute the product of two numbers? 1) Grid Method: Split both the factors into units, tens, hundreds etc, multiply all of these parts separately and then add them together. 2) Long Multiplication: Multiply the […]
Fractional Parts of log n
Let denote the natural logarithm. We consider the fractional parts as points on the interval . It is tempting to think that these points should spread uniformly around the interval, because : the sequence winds around the interval infinitely often, crossing each integer infinitely many times. But this intuition confuses two different facts. A sequence […]
The Rademacher–Menshov maximal estimate for exponential series
Let us begin with the basic problem. Given an exponential sum what information about the coefficients is enough to ensure that its partial sums actually converge at almost every point? The first condition one naturally encounters is square summability: Because the exponentials are orthogonal, this condition immediately implies that the partial sums are Cauchy in […]
The divisor function at consecutive integers
How can we establish the events for infinitely many ? https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S0025579300010743 Theorem: There are infinitely many integers for which . Indeed, for large , the number of such is at least of order . https://academic.oup.com/imrn/article-abstract/2011/7/1439/687263?redirectedFrom=fulltext https://projecteuclid.org/download/pdf_1/euclid.rmjm/1250126920
Bernoulli Numbers
Bernoulli Numbers: They are defined as the coefficient appearing in the following polynomial expansion of sums of integer powers. Therefore are the constants of integration (discrete integration!). To get from to , just multiply with and integrate and add a term can computed using the fact that Generating function: Thus we can remember the relations […]
Ramanujan 6-8-10 Identity
Let and be any numbers such that Then Let Change of variables gives with The identity now reduces to We have the following expressions that prove the identity.
Maxwell’s equations
is the electric field, is the magnetic field, is the charge density, is the current density, is the electric constant, is the magnetic constant, and is the speed of light. They say: Maxwell’s equations are not four unrelated facts. At first, they look like four separate laws: one about electric charge, one about magnetic fields, […]
Fourier Series Convergence
Consider the one-sided Fourier partial sums This is one of the first Fourier series in which several different notions of convergence visibly separate from one another. The coefficients satisfy but Thus the coefficient sequence belongs to , but not to . Square summability will give an limit. The failure of absolute summability means that uniform […]
Some formulae for Riemann Zeta
Formulae for Riemann Zeta function that allow you to see the relations between partial sums and the function value at an arbitrary complex number: Using these formulae, it’s easy to see that
Probabilistic Method: Lower Bounds for Ramsey Numbers
One of the earliest and most influential applications of the probabilistic method in combinatorics was given by Paul Erdős in his 1947 paper Some Remarks on the Theory of Graphs. The argument is famous not because it is long or technically complicated, but because it introduced a powerful new way of proving existence. Instead of […]
The Duffing oscillator
For the harmonic oscillator, the motion is almost magically simple. Confined to a quadratic potential , a displaced particle oscillates as a sine or cosine forever. The system possesses a single, intrinsic clock: its period is strictly independent of amplitude. But if we add even the simplest nonlinear correction, the story changes in a deep […]
Liouville type identity and Elementary Proofs of Modular Identities
Liouville’s Identity: Let be an even function on integers, we have the identity This is easy to prove once we know the identity! Just compare the number of time occurs on both sides. For instance, only even terms occur. occurs when on the LHS and the number of times it occurs is corresponds to divisors […]
Geodesics
A geodesic is the correct replacement for a straight line on a curved space. If a curve lies in ordinary Euclidean space, being straight means that its acceleration vanishes: But if the curve is constrained to lie on a curved surface, its ambient acceleration need not vanish. For example, a great circle on a sphere […]
Estermann’s evaluation of Gauss Sum
Let be an odd prime, and consider the quadratic Gauss sum This sum admits a second description in terms of the Legendre symbol. Each nonzero quadratic residue modulo occurs exactly twice among the numbers , while each nonresidue occurs zero times. Consequently, Using the character form we first compute its absolute value: The terms with […]
Schur’s Evaluation of the Sign of Gauss Sum
Consider the quadratic Gauss sum where is an odd prime. Writing the expression using the quadratic character modulo , and squaring gives Thus the magnitude and possible direction of are already known: The squaring argument leaves exactly one ambiguity: which sign occurs? There are several ways to settle this ambiguity. One can use Poisson summation, […]
A Proof of Fermat’s Sums of Squares Theorem using Triple Product Identity
We present a proof of Jacobi’s formula for representation number for sums of two squares due to Michael D. Hirschhorn Start with the Jacobi’s Triple Product identity Plugging for then for multiply by and we obtain Differentiating with and plugging we get, Divide by which equals to get We also have Therefore we established Plugging […]
Pell’s equation
Let be a positive non-square integer.Consider the Pell equation Let If solves the equation , so does In fact, we see that there is a single which generates all the possible solutions this way. Theorem: (Minimal solution generates all the solutions) If is the minimal element of with and then every element with if of […]
Triple Product Identity, Gauss
Here is a proof of the Jacobi’s Triple Product Identity due to Gauss: Consider Notice that Thus by repeated application of the above identity, we get And so we proved the identity Dividing this identity by where (Assume is even) and substituting , we get Taking , we get the triple product identity
Jacobi Triple Product Identity
Jacobi’s triple product is one of the basic identities in the theory of theta functions and -series. It converts a bilateral theta series, indexed by all integers, into an infinite product. In the form we shall use, it say: For and , we have The identity is powerful because the two sides look as though […]
The four-square theorem through Hermitian forms and reduction
The equation already suggests grouping the four real variables into two Gaussian integers. Put and . Then Thus Lagrange’s theorem says that every positive integer is represented by the binary Hermitian form on . This reformulation changes the point of view. Rather than seeking four integers directly, we study positive definite Hermitian forms over the […]
Fermat’s Two-Squares Theorem: Involutions, Indefinite Forms
Fermat’s theorem on sums of two squares says that an odd prime can be written as if and only if . The proofs of Heath-Brown and Zagier are striking because they prove this theorem by counting fixed points of involutions. The guiding principle is simple: when an involution acts on a finite set, all non-fixed […]
The Four-Square Theorem using Hurwitz Quaternions
The familiar proof of Fermat’s two-square theorem through Gaussian integers has a very satisfying shape. For a prime , one first finds a solution of . This produces the Gaussian ideal The fact that has class number one, or more concretely that it is a principal ideal domain, turns this ideal into one generated by […]
Lagrange’s four-square theorem
The four-square theorem states that every nonnegative integer can be written as a sum of four integer squares: For instance, At first this is surprising. Squares are sparse, and there are genuine congruence obstructions to representing every integer by two or three squares. Yet four squares suffice for every nonnegative integer. The proof combines two […]
Four Square Theorem: Descent Proof
Let Lagrange’s four-square theorem says that every positive integer can be written as for some integers . Equivalently, every positive integer is the squared length of an integer vector in . At first this looks like a direct equation-solving problem: given , find four integers whose squares add to it. The classical descent proof instead […]
Fermat’s Sum of Two Squares: Reduction Proofs
We prove Fermat’s two-squares theorem for primes. The theorem says that whenever a prime is congruent to one modulo four, it is a sum of two squares. The purpose of this exposition is not merely to collect five proofs. It is to explain how the first family of proofs is really organized around one object […]
Rademacher’s three-term reciprocity law
The ordinary Dedekind reciprocity law already has two apparently different explanations. One is geometric: a rational line cuts a lattice rectangle into two complementary regions, and the staircases along their common boundary cancel except at the endpoints. The other is analytic: the same staircase information is encoded by cotangent poles, and the residue theorem says […]
Dedekind Sums Reciprocity with Cotangent Sums
We have already seen two ways to understand Dedekind reciprocity. The Carlitz generating-function proof starts with a rational line and follows its finite lattice staircase: horizontal and vertical edges cancel in pairs, leaving only endpoint terms. The direct Bernoulli proof unfolds the staircase into a finite rectangle and then determines the answer by its variation […]
Rademacher’s reciprocity law for shifted Dedekind Sums
Let be coprime. We want to prove the classical reciprocity law where the Dedekind sum is defined by Here for nonintegral , and for integral . Let be coprime integers. We use the periodic first Bernoulli function defined for every real . In particular, when . This is slightly different from the usual sawtooth function […]
Dedekind Sums, Carlitz Identity, Reciprocity
Let be coprime integers. Begin with the line of rational slope , drawn from to . This small geometric object already contains several familiar reciprocity laws. Eisenstein’s lattice proof of quadratic reciprocity studies the lattice points on one side of this line and extracts only a parity from the count. Dedekind reciprocity comes from keeping […]
Quintic Equations
To understand how to solve the general quintic equation, one must first reconsider what it means to “solve” an equation at all. Consider the familiar analytic formulation of an ordinary radical: This formula says that an ordinary radical is produced in two stages. First, integrate the elementary differential Second, apply the exponential function. The Hermite–Kronecker–Brioschi […]
Irrationality of Zeta(3) (Beuker’s Proof)
We want to prove irrationality of This is the only odd positive integer which is known to be irrational. First proof of irrationality was by Apery who used constructed some very good rational approximations to zeta(3) using some recurrence relations. This proof has a lot of connections to hypergeometric functions, modular forms and many interesting […]
Irrationality of Zeta(3) using Modular Forms
The proof is based on the following lemma. Let be power series in Suppose that for any , the -th coefficient in the Taylor series of is rational and has denominator dividing where , are certain fixed positive integers and is the lowest common multiple of Suppose there exist real numbers such that has radius […]
Apery’s Proof of Irrationality of Zeta(3)
We want to prove the irrationality of . We will use the following remarkable formula to achieve that. In fact, we will prove that for any rational , we have The strategy to prove irrationality is simple. If you can approximate the number to well by rationals, then the number has to be irrational. Precisely […]
Erdős and Niven (1942)- Integrality of Harmonic Sums
THEOREM: There is only a finite number of integers for which oneor more of the elementary symmetric functions of is an integer. Proof: For small enough (), the k-th elementary symmetric function of the is less than For larger use existence of primes in short intervals to find a prime in the interval then and […]
Pi is irrational
Assume that Consider Note that are both integers. (Small derivatives at 0 vanish and larger derivatives cancel the denominators from the coefficients coming after taking derivatives. And then use symmetry of f for ) But Contradiction!
p-adic valuation of Harmonic sums
Look at the p adic-valuation of Harmonic sums. For 2-valuation the denominator contains a power of 2 which equals the highest power of 2 less than When you take common denominators and compute the numerator- you see that all terms except this power of 2 will gives even contributions and this terms gives an odd […]
List of Quadratic Reciprocity Proofs
http://www.rzuser.uni-heidelberg.de/~hb3/fchrono.html Many proofs of quadratic reciprocity are presented here. Theorema Fundamentale in Doctrina de Residuis Quadraticis. Gauss has proof by induction by reducing the problem of computing to Legendre symbol for a particular pair of primes to smaller numbers. Being a residue is captured by an equation, but what about non-residue? Gauss ingenious idea is […]
BBP Formula for Pi
This formula due to Bailey-Borwein-Plouffe is discovered by using integer relation algorithm PSLQ. They searched for integer relations between the quantities , and found the above relation. Finding the relation is the harder part, proving it is easy. Proof: Therefore, We get Similar formula for Quest for Pi: https://www.davidhbailey.com//dhbpapers/pi-quest.pdf PSLQ Algorithm: https://www.davidhbailey.com/dhbpapers/pslq-comp-alg.pdfBBP Formula: https://www.experimentalmath.info/bbp-codes/bbp-alg.pdf
Ramanujan Formula for Pi, WZ method
This can be seen as specialization of the following identity at Proof: We provide a proof by WZ method. We want to find such that If we have such a , then we have and we can see that will be a constant. Choice found by algorithms: Now So we are done. https://arxiv.org/pdf/math/9306213.pdf
Pi (approximations, formulae)
Pi: Polygon Approximations: Using perimeters of inscribed and circumscribed polygons: Let be the perimeters of regular sided polygon inscribed, circumscribed. We have Archimedes used 96 sided polygon to get: Leibniz formula: An accelerated series can be obtained by Euler transform: Nilakanta Series: Madhava: Machin’s formula: Proof: By using we get Essentially equivalent to the identity […]
Quartic Equations
The quartic formula becomes much less mysterious once one sees what problem it is trying to solve. For a cubic, Cardano found a way to write the unknown as a sum of two quantities, arranged so that the mixed terms combine into the required linear term. For a quartic, the basic aim is different: one […]
Cubic Equations
The cubic formula is less mysterious if one does not begin by trying to guess a root. Instead, one changes variables until the cubic has a form in which its nonlinear part can be split into two pieces. The key point is that all of Cardano’s formula, Viète’s substitution, the trigonometric solution, and Lagrange resolvents […]
Kürschâk and Nagel’s theorems (Erdos 1932)
Consider the familiar reciprocal sums None of the above quantities are integers.The first, second, and fourth cases all follow from one very elementary principle. One looks for a prime which occurs in one denominator more strongly than it occurs in every other denominator. After the fractions are put over a common denominator, every term except […]
Betrand Postulate : Erdos( 1932)
Bertrand’s postulate states that for every integer , there is a prime satisfying The statement is elementary, but it is remarkably strong: no matter how far one goes along the number line, one never encounters a multiplicative gap as large as a factor of containing no primes. Erdős’s proof (1932) of this fact is centered […]
Hölder’s inequality is repeated Cauchy–Schwarz
Cauchy–Schwarz and Hölder’s inequality are the basic tools for controlling the interaction, or correlation, of two functions. For finite sequences and , their correlation is measured by the inner product The triangle inequality reduces the problem to estimating Thus the central question is this: how can one control the total interaction using only separate information […]
Fundamental Theorem of Algebra: an algebraic proof
We will prove Fundamental Theorem of Algebra which says that every nonconstant polynomial with complex coefficients has a complex root. Equivalently, every polynomial with complex coefficients can be broken completely into linear factors. For example, a polynomial of degree can be written in the form where The theorem says that once we have added the […]
Fundamental Theorem of Algebra I
Let where . We shall prove that has exactly complex roots, counted with multiplicity. The central observation is that the degree- term governs the polynomial at large scale. When is large, every lower-degree monomial is smaller than by at least one factor of . Thus, on a sufficiently large circle, the curve traced by is […]
Fermat’s proof of descent for n=4
The exponent-four case of Fermat’s Last Theorem says that there are no nonzero integers satisfying Fermat’s original method proves something stronger and, in a sense, more natural: has no solution in positive integers. Once this stronger statement is known, the exponent-four case follows at once, because a hypothetical equation would be an equation of the […]
Euler’s Proof of Fermat’s theorem for n=3
We will prove that there do not exist nonzero integers such that Replacing by , this is the same as saying that has no nonzero integer solutions. The statement is the exponent-three case of Fermat’s Last Theorem. Euler’s proof is an early and striking example of a general mathematical strategy: begin with the ordinary factorization […]
.Morley’s Trisector Theorem
Morley’s Trisector Theorem states that for any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle. Its beauty lies in the surprising emergence of a perfectly regular shape from an arbitrary starting triangle Let’s first look at direct proof by trigonometric computations. For a with angles , we have […]
Bernoulli numbers, Umbral Calculus, Volkenborn integrals
Within the study of special functions and number theory, certain notational conventions occasionally arise that are so elegant and effective they appear to be a kind of magic. One of the most beautiful examples of this is the umbral calculus developed for Bernoulli numbers, where the indexed numbers are formally treated as powers of a […]
Zeta(2)
Basel Problem (1644) asks to find the exact value of the series Euler (1735) showed that At first glance, this is a problem about a list of numbers. Yet its answer, contains , a constant associated with circles, periodicity, and geometry. The surprise is not merely that the sum has a closed form. It is […]
One-Seventh Triangle and Routh’s Theorem
Certain geometric facts have an appealing simplicity that makes one hope for an equally simple picture-proof. The one-seventh triangle is a good example. Start with a triangle . Choose points on respectively, so that each chosen point divides its side in the ratio The three cevians form a smaller central triangle. Its area is exactly […]
Eisenstein’s Lattice Point Proof of Quadratic Reciprocity
The Law of Quadratic Reciprocity is one of the central results of classical number theory. Gauss famously called it the “Theorema Aureum,” or Golden Theorem. It reveals a hidden symmetry between two different modular worlds. At first glance, the question “Is a square modulo ?” seems unrelated to the question “Is a square modulo ?” […]
A Neighbourhood Of Infinity 