# Number Gossip

(Enter a number and I'll tell you everything you wanted to know about it but were afraid to ask.)

## Unique Properties of 1729

- 1729 is a Ramanujan number, also called a taxi-cab number, the smallest number representable in two ways as a sum of two cubes: 1729 = 1
^{3}+12^{3} = 9^{3}+10^{3}
- 1729 is the smallest number that is a pseudoprime simultaneously to bases 2, 3 and 5
- Web pages about 1729:
- 1729 is the smallest Carmichael number n such that n-1 is a perfect power
- 1729 is the smallest Carmichael number whose prime factors form an arithmetic progression
- 1729 is the product of the first three happy primes
- 1729 is the smallest Carmichael number such that its Euler totient function is a perfect square
- 1729 is the smallest phi-radical number: composite number n such that rad(φ(n)) = rad(n-1)

## Rare Properties of 1729

The composite integer n is a *Carmichael* number if b^{n-1} = 1 (mod n) for every integer b
which is relatively prime with n.

Carmichael numbers behave like prime numbers with respect to the most useful primality test, that is they pretend to be prime.

## Common Properties of 1729

A positive integer greater than 1 that is not prime is called *composite*.

Composite numbers are opposite to prime numbers.

The number n is *deficient* if the sum of all its positive divisors except itself is less than n.

Compare with perfect and abundant numbers.

A number is *odd* if it is not divisible by 2.

Numbers that are not odd are even. Compare with another pair -- evil and odious numbers.

The number n is *odious* if it has an odd number of 1's in its binary expansion.

Guess what evil numbers are.

A number is said to be *square-free* if its prime decomposition contains no repeated factors.