# The legend of the taxi-cab number and unknown facts about it.

## Carmichael number:

Here, again we are exposed to Fermat’s Little theorem, wonderful how Fermat pops up almost everywhere! Let’s have a look at a test that is applied to check if a number is prime or not and we are going to use Fermat’s little theorem. It says if you have a number, ‘a,’ and you raise it to the power of a number, ‘n,’ and you subtract the original number, ‘a,’ again and this resultant is divisible by n, p has to be be a prime! The mathematical expression for it is:

## Euler Pseudoprime:

An odd composite integer n is called an Euler pseudoprime to base a, if a and n are co-prime, and

## Sphenic Number:

A sphenic number is any number that can be expressed as the product of three prime numbers. 1729, too is a sphenic number, as 1729 can be expressed as the product of 7, 13, and 19, all of which are primes!

## Zeisel number:

Zeisel number, probably first introduced by Kevin Brown, while he was looking for numbers that when plugged into 2^{k-1}+k yielded prime numbers. Later 1885 was pointed out to be such a number by Helmut Zeisel. Soon it was discovered that 1885 has prime factors that follow:

## Loeschian number:

Numbers that can be expressed in the form x^2 + x.y + y^2 for integer x, y are called the Loeschian numbers. 1729 is the smallest number that can be expressed as the loeschian number in four different ways, where (x,y) can be: (25,23), (32,15), (37,8) and (40,3).

## Galactic Algorithm:

Named after Richard Lipton and Ken Regan, a galactic algorithm outperforms all algorithms for sufficiently large (VERY large) problems. Galactic algorithms have never used for terrestrial data as it is suitable for very large ones.

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