196

196 = 2² x 7².

196 is the smallest square that contains a multidigit prime number as a substring.

196 = 14² is a perfect square.

196 is the difference of two square numbers: 196 = 50² - 48².

The consecutive squares 13² = 169 and 14² = 196 use the same digits; 31² = 961 also uses the same digits.

196 has a unique representation as a sum of three squares: 196 = 4² + 6² + 12².

196 is a pentagonal pyramidal number and a heptagonal pyramidal number.

It is conjectured that 196 is the lowest Lychrel number. It is the lowest number not known to produce a palindrome by repeated reversal and addition. The algorithm has been tested to more than 246 million digits without finding a palindrome, but no one has ever proved that it will never produce one.

The Mercedes-Benz W196 was an entry in the 1954 and 1955 Formula One car racing series, winning 9 of 12 races.

Source: Number Gossip

289

289 = 17².

289 is the largest 3-digit square with increasing digits.

289 has a representation as a sum of two squares: 289 = 8² + 15².

289 is the hypotenuse of a primitive Pythagorean triple: 289² = 161² + 240².

289 is a centered octagonal number.

The Ford Mustang 289 was a high-performance version of the car in the 1960s.

Source: Number Gossip

576

576 = 2⁶ x 3².

576 has a unique representation as a sum of three squares: 576 = 8² + 16² + 16².

576 is the number of 4 x 4 Latin squares.

576 is the smallest square such that it remains a square when its leading digit is increased by 1: 676 = 26².

576 = (2 + 3 + 5 + 7 + 11)² - (2² + 3² + 5² + 7² + 11²).

New Balance produces a 576 line of fitness walking shoes.

Source: Number Gossip

100

100 = 2² x 5².

100 has a representation as a sum of two squares: 100 = 6² + 8².

100 is the sum of the first four cubes: 100 = 1³ + 2³ + 3³ + 4³. It is the smallest square that is the sum of consecutive cubes.

100 is the sum of the 9th and 10th triangular numbers: 100 = 45 + 55.

100 is the sum of the first 10 odd numbers: 100 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19.

The first three primes add to 10, and the first 3² primes add to 10² = 100.

The Romans wrote 100 as C. This is an abbreviation for the Latin word centum. A cent is a hundredth of a dollar, and a centime was a hundredth of a franc.

Source: Prime Curios!

36

36 = 2² x 3². It is a perfect square.

36 is 100100 in base 2 (binary). It is 44 in base 8.

36 has a unique representation as a sum of three squares: 36 = 2² + 4² + 4².

36 is the smallest number with four representations as a sum of two distinct primes: 36 = 5 + 31 = 7 + 29 = 13 + 23 = 17 + 19.

36 is the 8th triangular number (the sum of the first eight numbers). There are not many numbers that are both square and triangular. The first four are 1, 36, 1225, and 41616.

36 is the sum of the first three cubes: 1³ + 2³ + 3³.

36 is the smallest number to have nine different factors: 1, 2, 3, 4, 6, 9, 12, 18, and 36.

William Shakespeare wrote 36 plays.

Source: Phillips, R. 1994 Numbers: Facts, Figures and Fiction. Cambridge University Press.

400

400 = 2⁴ x 5².

400 = 20², a perfect square.

400 = 7⁰ + 7¹ + 7² + 7³. Hence, 400 is 1111 in base 7.

400 has a representation as a sum of two squares: 400 = 12² + 16².

A full circle is 360 degrees or 400 grads.

Years divisible by 400 are leap years, following adoption of the current Gregorian calendar.

Source: Numeropedia

676

676 = 2² x 13².

676 = 26², a perfect square.

676 is the smallest palindromic square whose square root is not palindromic.

676 has a representation as a sum of two squares: 676 = 10² + 24².

676 is a divisor of 99⁴ - 1.

676 is 1010100100 in base 2 (binary).

676 Melitta is an asteroid discovered in 1909.

Source: Number Gossip

25

25 = 5², a perfect square.

25 has a representation as a sum of two squares: 25 = 3² + 4². It is the smallest square that can be written as a sum of two squares.

25 is the hypotenuse of a primitive Pythagorean triple: 25² = 7² + 24².

25 is the sum of the first five odd numbers: 25 = 1 + 3 + 5 + 7 + 9.

25 = 3 + 4 + 5 + 6 + 7.

25 is an octahedral number.

Every square number is the sum of two triangular numbers: 25 = 10 + 15.

Pachisi, the Hindu word for 25, is one of the national games of India.

Source: Phillips, R. 1994. Numbers: Facts, Figures and Fiction. Cambridge University Press.

144

144 = 2⁴ x 3².

144² = 12², a perfect square. If you reverse the digits, 21² = 441.

144 has a unique representation as a sum of three squares: 144 = 4² + 8² + 8².

144 is a square number. Divide it by 4 and you get 36, which is also square. Does this happen with any other square numbers?

144 is the 12th Fibonacci number.

144 is 99 in base 15.

The interior angles in a regular decagon measure 144 degrees. In counting objects, 144 is called a gross (a dozen dozens).

Source: WolframAlpha

961

961 = 31². It is a perfect square.

961 is the largest square with strictly decreasing digits. It is also the largest prime power of a prime with strictly decreasing digits.

The digits of 961 can be rearranged to give two other squares: 169 = 13² and 196 = 14².

961 is 12321 in base 5 and 4241 in base 6.

961 beer is brewed in Beirut, Lebanon.

Source: Number Gossip

900

900 = 2² x 3² x 5².

900 is a square number: 30 x 30 = 900.

900 is the smallest square that is the sum of different primes by using all ten digits: 900 = 503 + 241 + 89 + 67 = 509 + 283 + 61 + 47.

900 has a base 5 representation (12100) that begins with its base 9 representation (1210). Its base 7 representation is 2424.

900 has a representation as a sum of two squares: 900 = 18² + 24².


900 is the sum (in degrees) of the internal angles of a heptagon.

Source: Prime Curios!

784

784 = 2⁴ x 7².

784 is a square number: 28 x 28 = 784.

784 is the square of the sum of the first seven integers: (1 + 2 + 3 + 4 + 5 + 6 + 7)² = 784.

784 is the sum of the first seven cubes: 1³ + 2³ + 3³ + 4³ + 5³ + 6³+ 7³ = 784.

784 is 1002001 in base 3 and 3344 in base 6.

Asteroid 784 Pickeringia was discovered in 1914.

Source: Numeropedia

484

484 = 2² x 11²

484 is a square number: 22 x 22 = 484.

484 is a palindrome in base 3 (12221) and in base 10.

484 has exactly nine divisors: 1, 2, 4, 11, 22, 44, 121, 242, 484.


Juliett 484 was a Soviet guided-missile submarine that eventually ended up on display at the Russian Sub Museum in Providence, R.I.

Source: The Positive Integers

256

256 = 2⁸. It is the smallest prime power of a prime power of a prime power of a prime.

256 is a square number: 16 x 16 = 256.

256 is the number of possible values that can be stored in a byte (8 bits).

256 is one of two fourth powers using the same digits: 4⁴ = 256 and 5⁴ = 625.

256 is the number of integer partitions of 29 into distinct parts.

256 is the number of drams in a pound.

Source: Numeropedia

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

1849

1849 = 43², a perfect square.

1849 is a centered octagonal number.

1849 is a perfect power in which neighboring digits are distinct.

1849 is a powerful number whose digit reversal is a semiprime: 9481 = 19 x 499.

Felix Klein (1849–1925) was born on April 25, 1849 (4/25/1849). Klein took particular delight in pointing out that each of the day (5²), month (2²), and year (43²) is the square of a prime number.


Source: The MacTutor History of Mathematics Archive
Image: Portrait Gallery

49

49 is a perfect square: 49 = 7² . It is the first square whose digits are square.

49 is the smallest number with the property that it and its neighbors are squareful. In other words, each of these numbers contains at least one square in its factorization.

48 = 2² x 2² x 3
49 = 7²
50 = 2 x 5²

49 is the smallest composite number that is not divisible by any Fibonacci number greater than 1.

49 has a unique representation as a sum of three squares: 49 = 2² + 3² + 6².

49 is the smallest number with exactly eight representations as a sum of three distinct primes: 49 = 3 + 5 + 41 = 3 + 17 + 29 = 5 + 7 + 37 = 5 + 13 + 31 = 7 + 11 + 31 = 7 + 13 + 29 = 7 + 19 + 23 = 13 + 17 + 19.

49 is the atomic number of the element indium.

Source: Number Gossip

169

169 is a square: 13 x 13 = 169. Note that, in reverse order, 961 = 31².

169 is the smallest three-digit square whose digits are increasing. It is also the smallest square that is a prime number when turned upside down (691).

169 is the sum of seven consecutive primes: 13 + 17 + 19 + 23 + 29 + 31 + 37 = 169.

Add the factorials of its digits: 1! + 6! = 9! = 363,601. Repeat the process with 363,601 to get 3! + 6! + 3! + 6! + 0! + 1! = 1454. Repeat the process to get 169.

169 is the smallest square that is the difference of two cubes: 8³ – 7³ = 169.

169 has a representation as a sum of two squares: 169 = 5² + 12².

169 is the hypotenuse of a primitive Pythagorean triple: 169² = 119² + 120².

The number of nonequivalent starting hands in Texas hold ‘em poker is 169.

Sources: Number Gossip; Wikipedia

19600

19600 = 140², a perfect square.

19600 = 2⁴ x 5² x 7².

19600 is one of only three numbers that are simultaneously square and tetrahedral. Tetrahedral numbers correspond to the number of balls in a triangular pyramid in which each side contains n + 1 balls. They are the sum of consecutive triangular numbers. 19600 is the 48th tetrahedral number.

19600 =

140²

= 1 + 3 + 6 + 10 + 15 + . . . + 1176.

The others are 1 = 1² = 1 and 4 = 2² = 1 + 3.

A triangular pyramid with side length 5 contains 35 spheres. Each layer represents one of the first five triangular numbers.

19600 has a representation as a sum of two squares: 19600 = 84² + 112².

A contestant in Nathan’s Famous Hot Dog Eating Contest in New York who consumes 66 hot dogs (the world record) would ingest a total of 19,600 calories.

Sources: Wolfram MathWorld; Wikipedia; CalorieLab

529

529 = 23². It is a perfect square.

529 is the 32rd lazy caterer number: the maximum number of pieces in which it is possible to divide a circle (pancake or pizza) for 32 cuts.

The concatenation of the consecutive numbers 528 and 529 forms a square number: 528529 = 727².

Pancake cut into seven pieces with three straight cuts.

When it comes to genuine computation, however, to seeing a self-important number like 529 and panicking when you divide it into 2,200, or realizing that, hey, it's the square of 23! well, that calls for a very different number system, one that is specific, symbolic and highly abstract.—Natalie Angier, New York Times, Sept. 15, 2008.

Section 529 of the IRS tax code organizes 529 plans to encourage saving for higher education.

Source: New York Times; The On-Line Encyclopedia of Integer Sequences