Mathematical series | Articles containing proofs | Theorems about prime numbers

Divergence of the sum of the reciprocals of the primes

The sum of the reciprocals of all prime numbers diverges; that is: This was proved by Leonhard Euler in 1737, and strengthens Euclid's 3rd-century-BC result that there are infinitely many prime numbers and Nicole Oresme's 14th-century proof of the divergence of the sum of the reciprocals of the integers (harmonic series). There are a variety of proofs of Euler's result, including a lower bound for the partial sums stating that for all natural numbers n. The double natural logarithm (log log) indicates that the divergence might be very slow, which is indeed the case. See Meissel–Mertens constant. (Wikipedia).

Divergence of the sum of the reciprocals of the primes
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MegaFavNumbers - 73,496,969,437, palindromic primes, programming squares

Some properties of the number 73,496,969,437. Links and details below: Comment thread from James Grime's video (amicable numbers conjecture): https://www.youtube.com/watch?v=R2eQVqdUQLI Matt Parker's tangent video (biggest tangent of a prime): https://www.youtube.com/watch?v=A7eJb8n8zAw

From playlist MegaFavNumbers

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How to find the zeros of a polynomial using the sum of two cubes

👉 Learn how to find the zeroes of a polynomial equation/expression involving the sum/difference of two cubes. Given a polynomial having the sum of two cubes, the polynomial can be factored as follows: a^3 + b^3 = (a + b)(a^2 - ab + b^2). Similarly, given a polynomial having the difference

From playlist Zeros of a Polynomial by Factoring

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Learn how to find the all the zeros of a polynomial with sum of two cubes

👉 Learn how to find the zeroes of a polynomial equation/expression involving the sum/difference of two cubes. Given a polynomial having the sum of two cubes, the polynomial can be factored as follows: a^3 + b^3 = (a + b)(a^2 - ab + b^2). Similarly, given a polynomial having the difference

From playlist Zeros of a Polynomial by Factoring

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What is the multiplicity of a zero?

👉 Learn about zeros and multiplicity. The zeroes of a polynomial expression are the values of x for which the graph of the function crosses the x-axis. They are the values of the variable for which the polynomial equals 0. The multiplicity of a zero of a polynomial expression is the number

From playlist Zeros and Multiplicity of Polynomials | Learn About

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Euler-Mascheroni III: Another NUCLEAR proof of the infinitude of primes

Follow the channel's Instagram: @whatthehectogon https://www.instagram.com/whatthehect... Check out these channels! Marching West (a DnD channel run by my friend Bill) https://www.youtube.com/channel/UCFNd... Twitter: @WestMarching https://twitter.com/WestMarching Instagram: @marchingwes

From playlist Analysis

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Euler-Mascheroni V: The Meissel-Mertens Constant

Channel social media: Instagram: @whatthehectogon https://www.instagram.com/whatthehectogon/ Twitter: @whatthehectogon https://twitter.com/whatthehectogon Check out my friend Bill's DnD channel: Marching West https://www.youtube.com/channel/UCFNd... Associated social media: Twitter: @Wes

From playlist Analysis

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The Reciprocal Prime Series (this proof should be taught in calculus!)

►Get my favorite calculator app for your phone or tablet: MAPLE CALCULATOR: https://www.maplesoft.com/products/maplecalculator/download.aspx?p=TC-9857 ►Check out MAPLE LEARN for your browser to do compute and graph math: https://www.maplesoft.com/products/learn/?p=TC-9857 ►Check out MapleS

From playlist Cool Math Series

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Euler-Mascheroni II: a NUCLEAR proof on the infinitude of primes

Follow the channel's Instagram: @whatthehectogon https://www.instagram.com/whatthehect... Check out these channels! Marching West (a DnD channel run by my friend Bill) https://www.youtube.com/channel/UCFNd... Twitter: @WestMarching https://twitter.com/WestMarching Instagram: @marchingwes

From playlist Analysis

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Euler-Mascheroni I: Three proofs that the Harmonic Series diverges!

Follow the channel's Instagram: @whatthehectogon https://www.instagram.com/whatthehect... Check out these channels! Marching West (a DnD channel run by my friend Bill) https://www.youtube.com/channel/UCFNdEg2zs7l5di7astgNmWA Twitter: @WestMarching https://twitter.com/WestMarching Instagr

From playlist Analysis

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Learn how and why multiplicity of a zero make sense

👉 Learn about zeros and multiplicity. The zeroes of a polynomial expression are the values of x for which the graph of the function crosses the x-axis. They are the values of the variable for which the polynomial equals 0. The multiplicity of a zero of a polynomial expression is the number

From playlist Zeros and Multiplicity of Polynomials | Learn About

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Solve and write the linear factorization using the sum of two cubes

👉 Learn how to find the zeroes of a polynomial equation/expression involving the sum/difference of two cubes. Given a polynomial having the sum of two cubes, the polynomial can be factored as follows: a^3 + b^3 = (a + b)(a^2 - ab + b^2). Similarly, given a polynomial having the difference

From playlist Zeros of a Polynomial by Factoring

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Find the zeros factoring vs square root method

👉 Learn about zeros and multiplicity. The zeroes of a polynomial expression are the values of x for which the graph of the function crosses the x-axis. They are the values of the variable for which the polynomial equals 0. The multiplicity of a zero of a polynomial expression is the number

From playlist Zeros and Multiplicity of Polynomials | Learn About

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Prime Reciprocal Series with @blackpenredpen (Oxford Maths Interview Question)

Steve from blackpenredpen answers a real Oxford University maths admissions interview question set by Oxford Mathematician (and interviewing tutor) Dr Tom Crawford. The question looks at the divergence of the sum of the reciprocals of the prime numbers, using the Fundamental Theorem of A

From playlist Interviews

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What do the zeros roots tell us of a polynomial

👉 Learn about zeros and multiplicity. The zeroes of a polynomial expression are the values of x for which the graph of the function crosses the x-axis. They are the values of the variable for which the polynomial equals 0. The multiplicity of a zero of a polynomial expression is the number

From playlist Zeros and Multiplicity of Polynomials | Learn About

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Calculus 2

Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. This course is about calculus 2

From playlist Calculus

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700 years of secrets of the Sum of Sums (paradoxical harmonic series)

Today's video is about the harmonic series 1+1/2+1/3+... . Apart from all the usual bits (done right and animated :) I've included a lot of the amazing properties of this prototypical infinite series that hardly anybody knows about. Enjoy, and if you are teaching this stuff, I hope you'l

From playlist Recent videos

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Find all the zeros using the difference of two cubes

👉 Learn how to find the zeroes of a polynomial equation/expression involving the sum/difference of two cubes. Given a polynomial having the sum of two cubes, the polynomial can be factored as follows: a^3 + b^3 = (a + b)(a^2 - ab + b^2). Similarly, given a polynomial having the difference

From playlist Zeros of a Polynomial by Factoring

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Calculus 3 Full Course

This course is about calculus 3 and the following topics have been presented in this course in very details. ⭐ Table of Contents ⭐ ⌨️ (0:05) Sequences ⌨️ (38:21) Infinite series ⌨️ (1:07:31) The divergence and integral test ⌨️ (1:24:07) Comparison test ⌨️ (1:48:00)

From playlist Calculus

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MinutePhysics Maths Trolling (Adding Past Infinity)

A response to the video "Adding past infinity" by MinutePhysics http://www.youtube.com/watch?v=kIq5CZlg8Rg Unlike convergent series, divergent series do not have limits (the sum of the first n terms, are not getting closer to a particular value as n increases). However, if we assume a d

From playlist My Maths Videos

Related pages

Square-free integer | Brun's theorem | Meissel–Mertens constant | Euler product | Convergent series | Limit comparison test | Exponential function | Integral test for convergence | Proof by contradiction | Integer | Fundamental theorem of arithmetic | Basel problem | Euclid | Series (mathematics) | Harmonic series (mathematics) | Prime number | Divergent series | Franz Mertens | Natural logarithm | Q.E.D. | List of sums of reciprocals | Paul Erdős | Cardinality | Leonhard Euler | Euclid's theorem | Riemann zeta function | Multiplicative inverse