# Tag: Number Theory

• ## Struggling with coin tossing paradox

I was led to this by the discussion on the non-random nature of prime numbers – as apparently it inspired one of the authors of the paper that noted this. I am struggling a bit with the maths of this, so hopefully writing it out might either help me grasp what is wrong with my […]

• ## The beauty of the Riemann sphere

Reading Elliptic Tales: Curves, Counting, and Number Theory today, I came across the Riemann Sphere – a thing of beauty: a closed surface of infinite extent. I will explain the maths of the sphere in a moment, but I am left wondering if one of two things apply: (a) We are terrible at teaching maths […]

• ## y^2 + x^2 = 1

This entry is based on the prologue for the book Elliptic Tales: Curves, Counting, and Number Theory (challenging but quite fun reading on the morning Tube commute!): is the familiar equation for the unit circle and in the prologue the authors show how a straight line with a rational slope intersects a circle at two […]

• ## Maths or physics ‘A’ level student? Then read this book…

The Language of Mathematics: making the invisible visible Although I want to warmly recommend this book, it is not what I expected when I started reading it – another popular explanation of maths that just might contain an insight or two. Instead it is much more like a tour d’horizon of a first year of […]

the smallest natural number that cannot be described by an English sentence of up to one thousand letters Such a number cannot exist – if we take the sentence above as a legitimate description of the number – as the sentence above both describes the number that cannot be described in less than one thousand […]

• ## Yitang Zhang latest…

A few weeks ago the world of maths was pleasantly shocked when a hitherto largely unnoticed mathematician, Yitang Zhang, demonstrated that there are infinitely many (pairs of primes) primes at most 70,000,000 apart – ie find a prime, any prime, no matter how large, and there is another two  at most 70,000,000 numbers away. Zhang’s […]

• ## A (partial) answer to my Goedelian conundrum?

Last week I puzzled over what seemed to me to be the hand waiving dismissal, by both Alan Turing and Douglas Hofstadter of what I saw as the problem of humans being able to write true statements that the formal systems employed by computers could not determine – the problem thrown up by Goedel’s Incompleteness […]

• ## How I discovered the fundamental theorem of arithmetic by chance

Actually, of course, I rediscovered it. I have been attempting to read, for the third time Douglas Hofstadter‘s celebrated Godel, Escher, Bach: I bought a copy in Washington DC in 2009 and loved it (though didn’t get very far before I put it down for some reason) but I have always struggled to get deeply […]

• ## Finding “new” primes

To accompany How to Solve it, I also bought How to Prove It: A Structured Approach which deals with the construction of proofs. I am puzzled, though, by its treatment of Euclid’s famous proof of the infinite order of the set of primes. Not because it gets the proof wrong – but because I do […]