692

692 = 2² x 173

692 is a number that has no digits in common with its cube (331373888).


In Microsoft Windows, error 692 signals a "hardware failure in port or attached device."

The U.S.S. Allen M. Sumner (DD-692), a destroyer, served in the U.S. Navy from 1943 to 1973.

1225

1225 is the product of the square of 5 and the square of 7 and hence is the square of 35. It is the product of the squares of two consecutive primes, 5 and 7.

1225 is a triangular square number; it is both a perfect square and a triangular number. 1225 is also a hexagonal number.

A representation of consecutive hexagonal numbers.

Source: Wolfram MathWorld

105

"Every number is built by multiplying prime numbers—105, for example, is three times five times seven. They are like the atoms of arithmetic—the hydrogen and oxygen of the world of numbers."—Marcus du Sautoy, The Guardian, Nov. 1, 2008.

105 is a triangular number. It is the sum of consecutive integers: 1 + 2 + 3 + . . . + 13 + 14 = 105.

105 is the sum of consecutive integers in four other ways: 12 + 13 + . . . + 17 + 18 = 15 + 16 + 17 + 18 + 19 + 20 = 19 + 20 + 21 + 22 + 23 = 34 + 35 + 36 = 52 + 53.

Paul Erdös conjectured that 105 is the largest number n such that the positive values of n - 2^k are all prime: 105 - 2 = 103, 105 - 4 = 101, 105 - 8 = 97; 105 - 16 = 89; 105 - 32 = 73; 105 - 64 = 41.

Source: Wolfram MathWorld

2929

2929 = 29 x 101

2929 is a square-free number because its prime decomposition contains no repeated factors.

Source: Number Gossip

1875

1875 = 3 x 5⁴

1875 is the smallest order for which there are 21 groups.


Source: What's Special About This Number?

416

416 = 2⁵ x 13

416 is the number of subsets of the 15th roots of unity that add up to a real number.


416 is the area code for the city of Toronto. It is one of the original 86 area codes from 1947.

Episode 416 (4th season, 16th episode) of the TV series Numb3rs was titled "Atomic No. 33" and included references to non-Newtonian fluids, the Riemann zeta function, and Bayesian network analysis.

Sources: What's Special About This Number?; The Math Behind Numb3rs

63

63 = 7 x 9

63 = 6² + 3³ = 2⁶ – 1

63 is the smallest number such that the common alphabetical value of its representation in Roman numerals is equal to itself: LXIII -- 12 + 24 + 9 + 9 + 9 = 63).

63 is a Harshad (or Niven) number; that is, it is divisible by the sum of its digits.

63 is the number of partially ordered sets of 5 elements.

Source: Number Gossip

145

145 = 5 x 29

145 = 1! + 4! + 5! The only other numbers that are the sum of the factorials of their digits are 1, 2, and 40,585. These numbers are known as factorions.

145 is the sum of two squares in two different ways: 145 = 12² + 1² = 8² + 9².

Source: Wolfram MathWorld

118

118 = 2 x 59

118 is the smallest number that can be written as the sum of four triples, whose products are all equal: 118 = 14 + 50 + 54 = 15 + 40 + 63 = 18 + 30 + 70 = 21 + 25 + 72. The product of each triple is 37,800.

118 is the smallest three-digit composite number, all of whose other digit permutations are prime numbers: 181 and 811 are both prime.


Source: Number Gossip

74

74 = 37 x 2

74 is a hungry number. The kth hungry number is the smallest number n such that 2ⁿ contains the first k digits of the decimal expansion of pi: 5, 17, 74, 144, 144, 2003, . . . . The first number is 5 because 2⁵ is 32, which contains 3, the first digit of pi. They were named hungry numbers because they try to eat as much "pi" as possible.


74 is the atomic number of tungsten.

Source: Number Gossip

56

56 is a pronic number because it is the product of two consecutive numbers: 7 x 8 = 56.

56 is the sum of the first six triangular numbers: 1 + 3 + .6 + 10 + 15 + 21 = 56. Hence, it is a tetrahedral number, the number of balls you can put in a triangular pyramid: 1, 4, 10, 20, 35, 56, 84, 120, . . .

56 is the sum of six consecutive prime numbers: 3 + 5 + 7 + 11 + 13 + 17 = 56.


56 men signed the Declaration of Independence in 1776, which led to the formation of the United States of America.

Source: Number Gossip

1111

1111 = 11 x 101

1111 is the sum of the digits of the first 100 primes.

Armistice Day, Nov. 11 (11/11), is the anniversary of the symbolic end of World War I in 1918, when the leaders of several countries signed an agreement on the 11th hour of the 11th day of the 11th month to cease hostilities.

Source: Number Gossip

77

77 is the 22nd discrete biprime and the first of the (7.q) family. Since both 7 and 11 are Gaussian primes, this means that 77 is a Blum integer.

It is the sum of three squares, 4² + 5² + 6², as well as the sum of the first eight prime numbers.

During World War II in Sweden at the border with Norway, 77 was used as a password, because the tricky pronunciation in Swedish made it easy to instantly discern whether the speaker was native Swedish, Norwegian, or German.

Source: Wikipedia

508

508 = 127 x 2²

508 is the sum for four consecutive prime numbers: 113 + 127 + 131 + 137 = 508.

508 is the area code for Worcester, Mass.

In the United States, section 508 of the Rehabilitation Act requires that Federal agencies give disabled employees and members of the public access to information that is comparable to the access available to others.

Sources: Wikipedia

85

85 = 5 x 17

85 is a Smith number because the sum of its digits is equal to sum of the digits in its prime factorization: 8 + 5 = 13 = 5 + 1 + 7.

85 is a centered triangular number. Such a number can be represented by a dot in the center, with all other dots surrounding the center in successive triangular layers. The first few triangular numbers are: 1, 4, 10, 19, 31, 46, 64, 85, 109, . . .

85 is the largest known n for which 1² + 2² + 3² + . . . + n² = 1 + 2 + 3 + . . . + m has a solution.

There are 85 ways to tie a tie, according to a book by Thomas Fink and Yong Mao.

Sources: Wolfram MathWorld ; What's Special About This Number?

538

538 = 2 x 269

538 is the 10th open meandric number: 1, 1, 2, 3, 8, 14, 42, 81, 262, 538, 1828, . . . .

The Electoral College consists of 538 popularly elected representatives who formally select the President and the ice President of the United States.

435

435 = 3 x 5 x 29

435 is a triangular number and a hexagonal number.

435 is the number of ordered partitions of 16 into distinct parts.

The United States House of Representatives has 435 members.

1103

1103 is a Sophie Germain prime. If both p and 2p + 1 are prime, then p is a Sophie Germain prime: 1103 and 2207 are both primes.

1103² = 1216609 and 3011² = 9066121.

1103 is the number of graphs with 9 vertices and 8 edges.

Source: Prime Curios!