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 […]
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
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 […]
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
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 […]
Estermann’s evaluation of Gauss Sum
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 […]
Schur’s Proof of Sign of Gauss Sum
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 […]
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 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 […]
Quintic Equations:
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 […]
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 […]
Cubic Equations
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 […]
Kürschâk and Nagel’s theorems (Erdos 1932)
None of the above quantities are integers. Proof: For the first expression, look at the largest prime- when we clear denominators, the denominator is divisible by this prime and numerator is not.For the second expression, if the is smaller then then the quantity is less than one, otherwise there will be a prime between $m$ […]
Betrand Postulate- Tchebyshev estimates – Erdos( 1932 -1)
Basic idea to approximately count primes is to look at and look at it prime factorization. To get access to primes between and , we need to look at the binomial coefficients Another way to see it: Use and sum over . Use the resulting expression to estimate the number of primes. (This is just […]
Bernoulli numbers, Faulhaber’s formula, Umbral Calculus, Volkenborn integrals
Consider the following defining formula for Bernoulli numbers in terms of sums of powers of integers: If we formally think of as this formula becomes The formal replacement of with has many nice properties. For instance, the generating series of Bernoulli numbers can be thought of as The Bernoulli polynomial can seen as Consider the […]
Zeta(2)
Basel Problem 1644: Find the value of Euler (1735) showed that Euler’s Proof: But Comparing the coefficients of we get, Fourier Proof: Consider on as a periodic function. Parseval gives Apostol’s Proof: To compute the integral using the change of coordinates to get Using we get The integral are computed below. Hence
Eisenstein Lattice Proof of Quadratic Reciprocity:
Let be distinct odd primes. Start with the following expression for Legendre symbol. which detects if is square modulo or not. By writing , and multiplying the quantities , it is easy to see that where the sum is over even integers . We used that is just a permutation of Therefore we get Now […]