Here is a very interesting formula for pi, discovered by David Bailey, Peter Borwein, and Simon Plouffe in 1995:Pi =...
Continue reading...number theory
Large Counterexample
A positive integer is said to be of even type if its factorization into primes has an even number of primes. Otherwise it is...
Continue reading...Sums of Two Squares
Which whole numbers are expressible as sums of two (integer) squares? Here’s a theorem that completely answers the question, due...
Continue reading...Pythagorean Triples
Which triples of whole numbers {a, b, c} satisfy a2 + b2 = c2 ? Such triples are called Pythagorean triples because they are integer...
Continue reading...Sums of Three and Four Squares
How many squares does it take to express every whole number as the sum of squares? We saw that two...
Continue reading...Products of Sums of Two Squares
Here’s a nice theorem due to Fibonacci, in 1202. Theorem. If integers N and M can each be written as the...
Continue reading...Fibonacci GCD’s, please
Fibonacci numbers exhibit striking patterns. Here’s one that may not be so obvious, but is striking when you see it....
Continue reading...Prime Number Theorem
Fix some number N. What fraction of the integers less than or equal to N are prime? Thinking about it,...
Continue reading...Sum of Prime Reciprocals
It is a well-known fact that the harmonic series (the sum of the reciprocals of the natural numbers) diverges. But what about...
Continue reading...e is irrational
If e were rational, then e = n/m for some integers m, n. So then 1/e = m/n. But the series expansion for 1/e is 1/e =...
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