# 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 144

- 144 is a dozen dozens, or one gross
- 144 is the only composite square in the Fibonacci sequence
- 144 is the smallest square such that there exists another square - 441 - with the same set of digits
- 144 is the smallest abundant Fibonacci number
- 144 is the largest sum-product integer - the integer that is equal to the sum of its digits times the product of its digits
- 144 is the smallest square perimeter of a primitive Pythagorean triangle
- 144 is the smallest square with internal digits also forming a non-zero square
- 144 is the smallest number such that its fifth power is the sum of fewer than five other fifth powers, thus disproving Euler's sum of powers conjecture
- 144 is the smallest three-digit number in which the sum of distinct digits of the number equals the sum of its distinct prime divisors (144 = 2
^{4}*3^{4} and 1 + 4 = 2 + 3); [Cramer] checked* 79 is the least positive integer requiring the maximum number of terms, namely 19, to be expressed as a sum of positive fourth powers

## Rare Properties of 144

*Fibonacci* numbers are numbers that form the Fibonacci sequence. The Fibonacci sequence is defined as starting with 1, 1 and then each next term is the sum of the two preceding ones.

Fibonacci numbers are very common in nature. For example, a pineapple has 8 spirals if you count one way, and 13 if you count the other way.

The k-th *hungry* number is the smallest number n such that 2^n contains the first k digits of the decimal expansion of pi.

They are named *hungry* numbers because they try to eat as much "pi" as possible.

The number n is a *square* if it is the square of an integer.

## Common Properties of 144

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

They are abundant above perfection, not to mention deficiency. See perfect and deficient numbers.

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

Composite numbers are opposite to prime numbers.

A number is *even* if it is divisible by 2.

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

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

Guess what odious numbers are.

An integer n is *powerful* if for every prime p dividing n, p^{2} also divides n.

How much power? They all can be written as a^{2} b^{3}.

The number n is *practical* if all numbers strictly less than n are sums of distinct divisors of n.