Let σ(n) be the sum of the positive divisors of n and let gcd(a, b) be the greatest common divisor of a and b. Form an n by n matrix M whose (i, j) entry is σ(gcd(i, j)). Then the…

Let σ(n) be the sum of the positive divisors of n and let gcd(a, b) be the greatest common divisor of a and b. Form an n by n matrix M whose (i, j) entry is σ(gcd(i, j)). Then the…

Mills’ constant is a number θ such that the integer part of θ raised to a power of 3 is always a prime. We’ll see if we can verify this computationally with SymPy. from sympy import floor, isprime from sympy.mpmath…

Fermat famously claimed to have a proof of his last theorem that he didn’t have room to write down. Mathematicians have speculated ever since what this proof must have been, though everyone is convinced the proof must have been wrong.…

I posted a couple prime-generating fractions on Google+ this weekend and said that I’d post an explanation later. Here it goes. The decimal expansion of 18966017 / 997002999 is .019 023 029 037 047 059 073 089 107 127 149…

Gelfands’s question asks whether there is a positive integer n such that the first digits of jn base 10 are all the same for j = 2, 3, 4, …, 9. (Thanks to @republicofmath for pointing out this problem.) This…

Suppose you want to know when your great-grandmother was born. You can’t find the year recorded anywhere. But you did discover an undated letter from her father that mentions her birth and one curious detail: the 13-year and 17-year cicadas…

A week ago I wrote about Perrin numbers, numbers Pn defined by a recurrence relation similar to Fibonacci numbers. If n is prime, Pn mod n = 0, and the converse is nearly always true. That is, if Pn mod…

The Perrin numbers have a definition analogous to Fibonacci numbers. Define P0 = 3, P1 = 0, and P2 = 2. Then for n > 2, define Pn+3 = Pn+1 + Pn+0. The Concrete Tetrahedron says It appears that n…

My youngest daughter asked me this morning whether you can find the number 2013 in the digits of pi. I said it must be possible, then wrote the following Python code to find where 2013 first appears. from mpmath import…

The famous Riemann hypothesis is equivalent to the following not-so-famous conjecture: For every N ≥ 100, | log( lcm(1, 2, …, N) ) – N | ≤ 2 log(N) √N. Here “lcm” stands for “least common multiple” and “log” means…

Does the base 10 expansion of 2^n always contain the digit 7 if n is large enough? As of 1994, this was an open question (page 196 here). I don’t know whether this has since been resolved. The following Python…

G. H. Hardy tells the following story about visiting his colleague Ramanujan. I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to…

There’s been a lot of buzz lately about Shinichi Mochizuki’s proposed proof of the ABC conjecture, a conjecture in number theory named after the variables used to state it. Rather than explaining the conjecture here, I recommend a blog post…

A number is called perfect if it is the sum of its proper divisors, i.e. all divisors less than itself. For example, 28 is perfect because 1 + 2 + 4 + 7 + 14 = 28. Amicable numbers are…

Binomial coefficients are hardly ever powers. That is, there are strong restrictions on when the equation has integer solutions for l ≥ 2. There are infinitely many solutions for k = 1 or 2. If k = 3 and l…