692

692 = 2² x 173.

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

692 has a representation as a sum of two squares: 692 = 4² + 16².

692 is a member of the sequence generated when you start with 1, then repeatedly multiply by 5 and reverse the digits: 1, 5, 52, 62, 13, . . . , 592, 692, 643, . . .


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 has a representation as a sum of two squares: 1225 = 21² + 28².

1225 is the smallest number with three representations as a sum of four positive cubes: 1225 = 1³ + 2³ + 6³ + 10³ = 3³ + 7³ + 7³ + 8³ = 4³ + 6³ + 6³ + 9³.

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.

Pere Marquette 1225 is a steam locomotive.

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 the 14th 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.

105 is 77 in base 14.

Space Shuttle Mission STS-105 was launched into space on Aug. 10, 2001.

Source: Wolfram MathWorld

2929

2929 = 29 x 101.

2929 has two representations as a sum of two squares: 2929 = 15² + 52² = 25² + 48².

2929 is the hypotenuse of two primitive Pythagorean triples: 2929² = 1560² + 2479² = 1679² + 2400².

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

2929 is a divisor of 100¹⁴ - 1.

2929 Entertainment is a media company with interests ranging from movie production to video games.

Source: Number Gossip

1875

1875 = 3 x 5⁴

1875 is an economical number.

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

The product of the prime factors of 1875 plus one is a perfect square: 3 x 5 + 1 = 16 = 4².

1875 is a number n such that n - 4, n - 2, n + 2, and n + 4 are prime: 1871, 1873, 1877, and 1879 are all prime.


French mathematician Henri Lebesgue was born June 28, 1875.

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 has a representation as a sum of two squares: 416 = 4² + 20².

416 is a number with exactly 12 divisors: 1, 2, 4, 8, 13, 16, 26, 32, 52, 104, 208, 416.


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 = 3² x 7.

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.

63 is 333 in base 4.

M63 is a galaxy, known as the Sunflower Galaxy, in the constellation Canes Venatici.

Source: Number Gossip