A Neighbourhood Of Infinity

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 […]

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 […]

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 […]

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 […]

http://www.crm.umontreal.ca/anatomie06/programme.htmlhttp://www.crm.umontreal.ca/anatomie06/pdf/GoldstonMontrealTalk1.PDF

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: […]

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 […]

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 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 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-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: […]

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 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 […]

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: 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 […]

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.

Electric Fields and Magnetic fields 1. Flux of electric field through surface is proportional to charge inside region bound by the surface. 2. Flux of magnetic field is zero. 3. Circulation of electric field around a loop is the negative rate of change of flux of the magnetic field out of surface bounded by the […]

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

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 […]

We show Esterman’s evaluation of the Gauss sum It’s easy to see by squaring that which implies that What about the sign? Both cases together can be written as To show that the sign is positive we will prove that the real part of the above expression is at least Start with Multiplying with we […]

Consider the quadratic Gauss sum It’s easy to see that which implies that But how do we determine the sign? There are many ways to prove the result (Poisson summation, theta series, Reciprocity by Contour integrals, Esterman evaluation etc). Here we show a linear algebra proof by Schur. Schur’s proof: Let be the matrix whose […]

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 […]

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 […]

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’s triple product identity): For and , we have Let Switching the terms and we find that Shifting the indices we find Now these functional equations with , we get Therefore In fact apply the functional relations again to get, Therefore has no poles. We have Writing we , and using the functional equation we […]

Bring-Jerrard Quintic Equations: Any general quintic can be reduced to Bring-Jerrard form using the Tschirnhaus type transformations: 1. Using the shifts , we can remove the term. 2. It’s also possible to remove the term if we use the substitution and choose so that satisfies Details: where We can make by these equations which give […]

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 […]

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 […]

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 […]

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 […]

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!

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 […]

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 […]

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

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: 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 […]

Cardano’s formula: Given Use shifts to eliminate the term. doesn’t have term if Therefore The equation reduces to the depressed cubic Substitute The equation becomes We want such that solve the quadratic equation To get all the roots, once has to choose the cube roots properly. The two choices of cube roots used above have […]