Unsolved problems in mathematics | Diophantine equations
The sum of four cubes problem asks whether every integer is the sum of four cubes of integers. It is conjectured the answer is affirmative, but this conjecture has been neither proved nor disproved. Some of the cubes may be negative numbers, in contrast to Waring's problem on sums of cubes, where they are required to be positive. The substitutions , , and in the identity lead to the identitywhich shows that every integer multiple of 6 is the sum of four cubes. (More generally, the same proof shows that every multiple of 6 in every ring is the sum of four cubes.) Since every integer is congruent to its own cube modulo 6, it follows that every rational integer is the sum of five cubes of integers. In 1966, V. A. Demjanenko proved that any integer that is congruent neither to 4 nor to −4 modulo 9 is the sum of four cubes of integers. For this, he used the following identities: andThese identities (and those derived from them by passing to opposites) immediately show that any integer which is congruent neither to 4 nor to −4 modulo 9 and is congruent neither to 2 nor to −2 modulo 18 is a sum of four cubes of rational integers. Using more subtle reasonings, Demjanenko proved that integers congruent to 2 or to −2 modulo 18 are also sums of four cubes of integers. The problem therefore only arises for integers congruent to 4 or to −4 modulo 9. One example is but it is not known if every such integer can be written as a sum of four cubes. (Wikipedia).
In this video, I give an explicit formula for the sum of cubes, and I show in particular why it’s the square of the sum of integers. It is really clever and neat, enjoy! Sum of squares: https://youtu.be/gVMEtOXdhs8 Subscribe to my channel: https://youtu.be/c/drpeyam
From playlist Cool proofs
Sum of integers squared explained
Explanation on deriving the equation. In arithmetic, we often come across the sum of n natural numbers. Sum of squares refers to the sum of the squares of numbers. It is basically the addition of squared numbers. Support my channel with this special custom merch! https://www.etsy.com/list
From playlist Math formulas, proofs, ideas explained
Ex 2: Factor a Sum or Difference of Cubes
This video provides and example of how to factor a sum or difference of cubes with common factors. Library: http://www.mathispower4u.com Search: http://www.mathispower4u.wordpress.com
From playlist Factoring a Sum or Difference of Cubes
Picturing the Sum of Two Cubes
A geometric representation of the sum of two cubes.
From playlist Algebra
(New Version Available) Factoring a Sum or Difference of Cubes
New Version: https://youtu.be/pRgiZ9pLnOc The video explains how to fact polynomials in the form a^3 + b^3 and a^3 + b^3. http://mathispower4u.wordpress.com/
From playlist Factoring a Sum or Difference of Cubes
Guess and Check? Rediscover the sum of cubes formula for the first n natural numbers
We know that sum of cubes of first n natural numbers is = (n(n+1)/2)2. Is it possible to discover this fact by guess and check after you study the sum of first n integers and the sum of squares of the first n integers: we assume sum of cubes is a degree 5 polynomial in n. then we use 5 eq
From playlist Elementary Number Theory
Lesson: Factoring a Sum or Difference of Cubes
This video verifies for factoring formulas and provides examples on how to factor a sum or difference of cubes. http://mathispower4u.com
From playlist Factoring a Sum or Difference of Cubes
Find a Partial Sum Using Summation Formula: Sum(5i^3-2i)
This video explains how to determine a partial sum given in sigma notation using summation formulas. http://mathispower4u.com
From playlist Series (Algebra)
Theory of numbers: Congruences: Introduction
This lecture is part of an online undergraduate course on the theory of numbers. This lecture introduces congruences. We give some examples of using congruences to study the problem of which integers can be written as a sum of 2 or 3 squares or 3 cubes. For the other lectures in the
From playlist Theory of numbers
Factoring Polynomials Sum Difference of Cubes
I explain how to factor the Sum of Cubes and Difference of Cubes patterns and then work through many examples to help you understand the process. These are a special type of Polynomials. Here you can find a selection of my Factoring Lessons http://www.profrobbob.com/algebra-2/factoring-te
From playlist Algebra 2
Definite Integrals with the Limit Definition - Full Tutorial
This is a complete introduction to definite integrals. It starts with the familiar formulas needed, then goes on to a formal construction of the definite integral using Riemann Sums. Several examples of finding definite integrals with the limit definition are given. If you enjoyed this vi
From playlist Math Tutorials
2021 MATHCOUNTS State Live Solve - 2 by Po-Shen Loh
Welcome to the Official YouTube Channel of the Daily Challenge with Po-Shen Loh! Please subscribe to stay in touch. This time, Prof. Po-Shen Loh is going to collaborate with MATHCOUNTS, one of the biggest middle school math competitions in the US, AGAIN! Prof. Loh is going to solve all p
From playlist MATHCOUNTS
Ex 1: Factor a Sum or Difference of Cubes
This video provides and example of how to factor a sum or difference of cubes. Library: http://www.mathispower4u.com Search: http://www.mathispower4u.wordpress.com
From playlist Factoring a Sum or Difference of Cubes
A tricky fraction question from India!
Thanks to Tanmay for the suggestion! And do check out Tanmay's channel Mr. Whiteboard for mathematics videos in Hindi: https://www.youtube.com/channel/UCNBjkDt8Xxn9I4Nb_Cn6ZoA Thanks to Tony H. and Nestor for pointing out mistakes in an earlier version of this video! This question is ada
From playlist Math Puzzles, Riddles And Brain Teasers
Factoring Polynomials Completely - All Types (100 Problems & Free Worksheet)
Learn how to factor polynomials completely in this video math tutorial by Mario's Math Tutoring. We will be going through 100 factoring problems including greatest common factor, difference of 2 squares, sum and difference of 2 cubes, perfect square trinomials, trinomials with leading coe
From playlist Factoring - All Different Types
The Uncracked Problem with 33 - Numberphile
Update March 2019: 8866128975287528^3+(-8778405442862239)^3+(-2736111468807040)^3 33 is the lowest unsolved problem in the world of "summing three cubes". But 30 was also a tough nut to crack! More links & stuff in full description below ↓↓↓ Featuring Tim Browning from the University of
From playlist Sum of Three Cubes - Numberphile
LMS Popular Lecture Series 2013, Addictive Number Theory
Addictive Number Theory by Dr Vicky Neale Held at the Institute of Education in London
From playlist LMS Popular Lectures 2007 - present
Sums of Two Cubes by Ari Shnidman
PROGRAM ELLIPTIC CURVES AND THE SPECIAL VALUES OF L-FUNCTIONS (HYBRID) ORGANIZERS: Ashay Burungale (CalTech/UT Austin, USA), Haruzo Hida (UCLA), Somnath Jha (IIT Kanpur) and Ye Tian (MCM, CAS) DATE: 08 August 2022 to 19 August 2022 VENUE: Ramanujan Lecture Hall and online The program pla
From playlist ELLIPTIC CURVES AND THE SPECIAL VALUES OF L-FUNCTIONS (2022)
A slice or two of a diagonal cubic: arithmetic stratification via the circle method - Trevor Wooley
Joint IAS/Princeton University Number Theory Seminar Topic: A slice or two of a diagonal cubic: arithmetic stratification via the circle method Speaker: Trevor Wooley Affiliation: Purdue University Date: October 24, 2019 For more video please visit http://video.ias.edu
From playlist Mathematics
Sum of Two Cubes - Understanding the Formula
Understanding why the sum of two cubes is factored into its common formula representation
From playlist Algebra