In number theory, a Liouville number is a real number x with the property that, for every positive integer n, there exist infinitely many pairs of integers (p, q) with q > 1 such that

Liouville numbers are "almost rational", and can thus be approximated "quite closely" by sequences of rational numbers. They are precisely the transcendental numbers that can be more closely approximated by rational numbers than any algebraic irrational number. In 1844, Joseph Liouville showed that all Liouville numbers are transcendental, thus establishing the existence of transcendental numbers for the first time.

]]>0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666...

The triangle numbers are given by the following explicit formulas:

where is a binomial coefficient. It represents the number of distinct pairs that can be selected from n + 1 objects, and it is read aloud as "n plus one choose two".

14. **Mersenne prime**: In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form

The exponents n which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ...

Numbers of the form

without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that n be prime. The smallest composite Mersenne number with prime exponent n isMersenne primes were studied in antiquity because of their close connection to perfect numbers: the Euclidâ€“Euler theorem asserts a one-to-one correspondence between even perfect numbers and Mersenne primes.

As of July 2020, 51 Mersenne primes are known. The largest known prime number,

, is a Mersenne prime. Since 1997, all newly found Mersenne primes have been discovered by the Great Internet Mersenne Prime Search, a distributed computing project.]]>In number theory, odious numbers are numbers that have an odd number of digits in their binary expansion. They determine the locations of the non-zero integers in the Thue-Morse sequence.

Some examples:

1 (1)

4 (100)

5 (101)

6 (110)

7 (111)

12. **Evil Numbers**

Evil numbers are the opposite of odious numbers. They have an even number of digits in their binary expansion and determine the locations of the zeroes in the Thue-Morse sequence.

Some examples:

2 (10)

3 (11)

8 (1000)

9 (1001)

10 (1010)

2. Amicable numbers: Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number.

The smallest pair of amicable numbers is (220, 284). They are amicable because the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71 and 142, of which the sum is 220. (A proper divisor of a number is a positive factor of that number other than the number itself. For example, the proper divisors of 6 are 1, 2, and 3.)

3. Abundant number: An abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number itself. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance. The number 12 has an abundance of 4, for example.

4. Deficient number: A number that is greater than the sum of all of its divisors except itself. The factors of 22 are 1, 2 and 11 and 22, and 1 + 2 + 11 = 14, which is less than 22, so 22 is a deficient number.

5. Happy Number: A happy number is defined by the following process:

Starting with any positive integer, replace the number by the sum of the squares of its digits in base-ten, and repeat the process until the number either equals 1 (where it will stay), or it loops endlessly in a cycle that does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers (or sad numbers).

6. Sad number: An unhappy number is a number that is not happy, i.e., a number such that iterating this sum-of-squared-digits map starting with. never reaches the number 1. The first few unhappy numbers are 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 15, 16, 17, 18, 20, ...

7. Taxicab number: In mathematics, the nth taxicab number, typically denoted Ta(n) or Taxicab(n), also called the nth Hardyâ€“Ramanujan number, is defined as the smallest integer that can be expressed as a sum of two positive integer cubes in n distinct ways. The most famous taxicab number is 1729 = Ta(2) =

The name is derived from a conversation in about 1919 involving mathematicians G. H. Hardy and Srinivasa Ramanujan. As told by Hardy:

I remember once going to see him [Ramanujan] when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two [positive] cubes in two different ways."

8. Complex number: A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation

Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, and b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.Complex numbers allow solutions to certain equations that have no solutions in real numbers. For example, the equation

has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem. The idea is to extend the real numbers with an indeterminate i (sometimes called the imaginary unit) that is taken to satisfy the relation

so that solutions to equations like the preceding one can be found.9. Transcendental number: In mathematics, a transcendental number is a number that is not algebraic - that is, not a root (i.e., solution) of a nonzero polynomial equation with integer or equivalently rational coefficients. The most popular transcendental numbers are

10. In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0,

where the Greek letter phi or represents the golden ratio. It is an irrational number that is a solution to the quadratic equation , with a value of:]]>