The terms of the two integer sequences below are equal for all 
Posts Tagged 'Number Theory'
A Remarkable Pair of Sequences
Published June 8, 2017 Occasional Leave a CommentTags: Number Theory
A Geometric Sieve for the Prime Numbers
Published April 27, 2017 Occasional Leave a CommentTags: Number Theory, Primes
In the time before computers (BC) various ingenious devices were invented for aiding the extensive calculations required in astronomy, navigation and commerce. In addition to calculators and logarithms, several nomograms were devised for specific applications, for example in meteorology and surveying.
Numerical Coincidences
Published March 23, 2017 Occasional Leave a CommentTags: Number Theory, Recreational Maths
A numerical coincidence is an equality or near-equality between different mathematical quantities which has no known theoretical explanation. Sometimes such equalities remain mysterious and intriguing, and sometimes theory advances to the point where they can be explained and are no longer regarded as surprising.
Cosine of 355 radians is almost exactly equal to -1. Is this a coincidence? Read on!
Brun’s Constant and the Pentium Bug
Published March 9, 2017 Occasional Leave a CommentTags: Arithmetic, Euler, Number Theory
Euclid showed by a deliciously simple argument that the number of primes is infinite. In a completely different manner, Euler confirmed the same result. Euler’s conclusion followed from his demonstration that the sum of the reciprocals of the primes diverges:
Obviously, this could not happen if there were only finitely many primes.
The Shaky Foundations of Mathematics
Published December 1, 2016 Irish Times Leave a CommentTags: Algorithms, Logic, Number Theory
The claim is often made that mathematical results are immutable. Once proven, they remain forever valid. But things are not so simple. There are problems at the very core of mathematics that cast a shadow of uncertainty. We can never be absolutely sure that the foundations of our subject are rock-solid [TM104 or search for “thatsmaths” at irishtimes.com].
Left: Plato and Aristotle. Centre: Pythagoras. Right: Euclid [Raphael, The School of Athens]
The ancient Greeks put geometry on a firm footing. Euclid set down a list of axioms, or basic intuitive assumptions. Upon these, the entire edifice of Euclidean geometry is constructed. This axiomatic approach has been the model for mathematics ever since.
Every number is interesting. Suppose there were uninteresting numbers. Then there would be a smallest one. But this is an interesting property, contradicting the supposition. By reductio ad absurdum, there are none!
This is the hundredth “That’s Maths” article to appear in The Irish Times [TM100, or search for “thatsmaths” at irishtimes.com]. To celebrate the event, we have composed an ode to the number 100.
The counting numbers that we learn as children are so familiar that using them is second nature. They bear the appropriate name natural numbers. From then on, names of numbers become less and less apposite.
