Advice for Maths Exploration | Chebyshev and Spread Polynumbers: the remarkable Goh factorization

A key challenge for amateur mathematicians is finding a fruitful and accessible and interesting area for investigation. This is not so easy: classical number theory is certainly very interesting but it is highly difficult, perhaps even unrealistic, to hope to make really new discoveries here at the amateur, elementary level. Not impossible, but do you want to spend a huge amount of time and effort for little probable gain?
A good alternative to classical Number theory is "Poly Number theory" which concerns itself with the rich algebra of polynomials / polynumbers, in which interesting families feature prominently -- like the Chebyshev polynomials of the first and second kinds, and their more modern variants -- the spread polynomials, which arise in Rational Trigonometry as a natural replacement to the Chebyshev polynomials of the first kind. My book "Divine Proportions: Rational Trigonometry to Universal Geometry" has an entire chapter on these (available at wildegg.com).
In this video we introduce these families, with particular emphasis on the spread polynumbers and their remarkable factorization, which sets them clearly apart from the Chebyshev polynomials and is a strong indication that they are really much more fundamental!!
There is a lot of number theory that emerges from this factorization, which was found by Shuxiang Goh about 15 years ago while an undergraduate at UNSW (he is now a practicing medical doctor).
Along the way, I mention why the famed "Fundamental Theorem of Algebra" is in fact a fiction: it is only approximately correct, not exactly correct, and it really is crucial for modern undergraduates to appreciate this. Factorization of polynomials / polynumbers is a MUCH more interesting topic than it is currently perceived to be.
And I also stress that identities involving "cos x" and "sin x" etc should NOT be viewed in terms of "equality of functions", but rather more concretely in terms of equality of coefficients in a power series / polyseries. This makes much more sense, and is actually doable! So there are advantages in thinking more clearly about foundational issues in mathematics --- we can connect better with modern computation, and new theoretical doors open.
This video is part of a series on Advice to Prospective research mathematicians at my sister channel Wild Egg Maths, available to Members there (see below).
A big thanks to all my Subscribers, Members of the Wild Egg Maths channel, and Patreon supporters!
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Here are all the Insights into Mathematics Playlists:
Elementary Mathematics (K-6) Explained: / playlist
list=PL8403C2F0C89B1333
Year 9 Maths: • Year9Maths
Ancient Mathematics: • Ancient Mathematics
Wild West Banking: • Wild West Banking
Sociology and Pure Mathematics: • Sociology and Pure Mat...
Old Babylonian Mathematics (with Daniel Mansfield): / playlist
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: • MathHistory: A course ...
Wild Trig: Intro to Rational Trigonometry: • WildTrig: Intro to Rat...
MathFoundations: • Math Foundations
Wild Linear Algebra: • Wild Linear Algebra
Famous Math Problems: • Famous Math Problems
Probability and Statistics: An Introduction: • Probability and Statis...
Boole's Logic and Circuit Analysis: • Boole's Logic and Circ...
Universal Hyperbolic Geometry: • Universal Hyperbolic G...
Differential Geometry: • Differential Geometry
Algebraic Topology: • Algebraic Topology
Math Seminars: • MathSeminars
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And here are the Wild Egg Maths Playlists:
Solving Polynomial Equations: • Solving Polynomial Equ...
Exceptional Structures in Maths and Physics via Dynamics on Graphs" • Dynamics on Graphs (Me...
Triangle Centres: • ENCYCLOPEDIA OF TRIANG...
Six: An elementary course in pure mathematics: • Six: An elementary cou...
Algebraic Calculus One: • Algebraic Calculus One
Algebraic Calculus Two: • Algebraic Calculus Two