# 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 29

- The 29th power of two is the largest power of two to have all different digits
- 29 is the smallest multi-digit prime whose product of digits of its cube is also a cube
- 29 is the smallest multi-digit number which on adding its reverse gives a perfect square (i.e., 29 + 92 = 121 = 11
^{2})
- 29 is the largest number such that sum of its digits is greater than the number of primes up to it
- 29 is the smallest prime of form p
^{p} + 2
- 29 is the smallest prime equal to the sum of three consecutive squares: 29 = 2
^{2} + 3^{2} + 4^{2}
- There are 29 distinct three-dimensional pentacubes (glued from five identical cubes by joining faces)

## Common Properties of 29

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.

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

Guess what odious numbers are.

The n-th *lazy caterer* number is the maximum number of pieces a (circular) pizza can be cut into with n (straight-line) cuts.

Unlike the situation with cake, everybody gets the toppings.

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.

A *prime* is a positive integer greater than 1 that is divisible by no positive integers other than 1 and itself.

Prime numbers are opposite to composite numbers.

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

A prime number is called a *twin* prime if there exists another prime number differing from it by 2.