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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 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 […]

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 […]

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 […]

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 […]

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 […]