Prime Time π

Monday, April 30, 2018

Happy Birthday to Gauss, Father of the First Predictive Algorithm

Today, April 30th, we celebrate the birthday of Carl Friedrich Gauss, often called the “Prince of Mathematicians.”

There are some fun details about his life and work available via the Google doodle page:

Image source: Google Doodle of the Day

The Quartz site also has an excellent article: https://qz.com/1265493/carl-friedrich-gauss-celebrating-the-father-of-the-predictive-algorithm

Thursday, January 4, 2018

New Largest Prime - 23,249,425 Digits!

The Great Internet Mersenne Prime Search (GIMPS) has discovered the largest known prime number, 2^77,232,917 - 1, having 23,249,425 digits. The number, called M77232917, is the 50th "Mersenne prime" to be discovered. https://www.mersenne.org/primes/press/M77232917.html

Monday, October 16, 2017

Almost All Numbers Contain the Digit Three

My mind was blown today when I came across the assertion (and proof!) that almost all numbers contain the digit '3'.

For small numbers of a given length of digits, you can count the ones that contain the digit '3'. You can see these values in the following table:

Number of digits	Examples	% Containing 3
1 (0-9)	3	10% (1)
2 (0-99)	3,13,23,30,31,32,33,34,35,36,37,38,39,43,53,...	19% (19)
3 (0-999)	3,13,...,30-39,...,300-399,430-439,...	27% (271)

You can count them by hand, but the count 'C' for any given number of digits 'n' is given by the following formula:

C = 10^n - 9^n

Thus the percentage is given by:

% = (10^n - 9^n) / 10^n
% = (10^n / 10^n) - (9^n / 10^n)
% = 1 - (9/10)^n

Taking the limit of this a n -> inf and you see that % = 1.

James Grimes does an excellent job of explaining it in this video:
https://www.youtube.com/watch?v=UfEiJJGv4CE

Monday, July 10, 2017

Palindrome Dates

If you follow the traditional American format of mm/dd/yyyy for dates, then today, 7/10/2017 is "palindromic". The next date to follow this format will be 8/10/2018. Coincidentally, it also works for the mm/dd/yy format -- 7/10/17. If you follow the European format of dd/mm/yyyy, you're going to have to wait until October 7, 2017 -- which, you guessed it -- also comes out as 7/10/2017. PS. Yes, I know I've played fast and loose with leading zeroes.

Thursday, July 6, 2017

Climb To Prime

John H. Conway’s proposed a list of math problems for which he is offering $1000 for a solution. One of those is his “Climb to a Prime” conjecture, which is stated as:

Problem 5. Climb to a Prime:

Let n be a positive integer. Write the prime factorization in the usual way, e.g. 60 = 2² · 3 · 5, in which the primes are written in increasing order, and exponents of 1 are omitted. Then bring exponents down to the line and omit all multiplication signs, obtaining a number f(n). Now repeat.

So, for example, f(60) = f(2² · 3 · 5) = 2235. Next, because 2235 = 3 · 5 · 149, it maps, under f, to 35149, and since 35149 is prime, it maps to itself. Thus 60 → 2235 → 35149 → 35149 → ..., so we have climbed to a prime, and we stop there forever. The conjecture, in which I seem to be the only believer, is that every number eventually climbs to a prime. The number 20 has not been verified to do so. Observe that 20 → 225 → 3252 → 223271 → ... , eventually getting to more than one hundred digits without yet reaching a prime!

James Davis took up the challenge and found a counterexample: 13532385396179 = 13 ⋅ 53² ⋅ 3853
This prime number fails the conjecture because its factorization maps back to itself!

You can read a few more details here:

http://www.popularmechanics.com/science/math/news/a26815/why-13532385396179-is-a-magic-number/