Terence Tao has uploaded his slides on the Collatz conjecture (targeted at high school students): https://terrytao.files.wordpress.com/2020/02/collatz.pdf
A very enjoyable read indeed.
The best “encyclopedic” reference on the Collatz conjecture is the one listed below, published by the American Math Society. Note that the Collatz conjecture remains unsolved as of today.
These are two excellent videos explaining Holder’s Inequality for Lp Spaces:
Featured book:
Divine Proportions: Rational Trigonometry to Universal Geometry
This revolutionary book establishes new foundations for trigonometry and Euclidean geometry. It shows how to replace transcendental trig functions with high school arithmetic and algebra to dramatically simplify the subject, increase accuracy in practical problems, and allow metrical geometry to be systematically developed over a general field. This new theory brings together geometry, algebra and number theory and sets out new directions for algebraic geometry, combinatorics, special functions and computer graphics. The treatment is careful and precise, with over one hundred theorems and 170 diagrams, and is meant for a mathematically mature audience. Gifted high school students will find most of the material accessible, although a few chapters require calculus. Applications include surveying and engineering problems, Platonic solids, spherical and cylindrical coordinate systems, and selected physics problems, such as projectile motion and Snell’s law. Examples over finite fields are also included.
The Riemann Zeta function is an extremely important special function of mathematics and physics. It is intimately related with very deep results surrounding the prime numbers. Now why would we want to care about prime numbers? Well, the entire concept of web security is built around prime numbers. Most of the algorithms for banking security, cryptography, networking, communication, etc are constructed using these prime numbers and the related theorems. The reason we do this is because of the inherently sporadic nature of prime numbers. You never know where the next one is going to appear on the number line! So what does that have to do with the Reimann zeta function? If prime numbers are random, what’s the point of looking into them?
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Source: http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss
Johann Carl Friedrich Gauss (/ɡaʊs/; German: Gauß, pronounced [ɡaʊs] (
listen); Latin: Carolus Fridericus Gauss) (30 April 1777 – 23 February 1855) was a German mathematician and physical scientist who contributed significantly to many fields, including number theory, algebra, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.
Sometimes referred to as the Princeps mathematicorum[1] (Latin, “the Prince of Mathematicians” or “the foremost of mathematicians”) and “greatest mathematician since antiquity“, Gauss had a remarkable influence in many fields of mathematics and science and is ranked as one of history’s most influential mathematicians.[2]
Continue reading at http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss
*Not Einstein!
Ferdinand Gotthold Max Eisenstein (16 April 1823 – 11 October 1852) was a German mathematician. He specialized in number theory and analysis, and proved several results that eluded even Gauss. Like Galois and Abel before him, Eisenstein died before the age of 30. He was born and died in Berlin, Prussia.
Gauss … in conversation once remarked that, there had been only three epoch-making mathematicians: Archimedes, Newton, and Eisenstein.
Source: http://en.wikipedia.org/wiki/Gotthold_Eisenstein
Source: http://www.artofproblemsolving.com/Resources/Papers/SatoNT.pdf
Excellent notes on Olympiad Number Theory!
Preface:
This set of notes on number theory was originally written in 1995 for students
at the IMO level. It covers the basic background material that an IMO
student should be familiar with. This text is meant to be a reference, and
not a replacement but rather a supplement to a number theory textbook;
several are given at the back. Proofs are given when appropriate, or when
they illustrate some insight or important idea. The problems are culled from
various sources, many from actual contests and olympiads, and in general
are very difficult. The author welcomes any corrections or suggestions.
BBC Horizon programme. Simon Singh’s moving documentary of Andrew Wiles’ extraordinary search for the most elusive proof in number theory.
Singapore Maths Tuition 