521

521 is a prime number.

521 has a representation as a sum of two squares: 521 = 11² + 20².

521 is the 13th Lucas number.

521 and 523 for a twin prime pair.

521 is the hypotenuse of a primitive Pythagorean triple: 521² = 279² + 440².

521 has the representation 521 = 2⁹ + 9.

521 is the exponent of the Mersenne prime 2⁵²¹ - 1.

521 is the smallest prime whose reversal is a cube: 125 = 5³.

Galaxy cluster Abell 521 is located about 2.9 billion light-years from Earth.

Source: Prime Curios!

503

503 is a prime number.

503 is the smallest prime that is the sum of the cubes of the first n primes: 2³ + 3³ + 5³ +7³ = 503.

5! + 0! + 3! is a three-digit Mersenne prime.

503 is a number that cannot be written as a sum of three squares.


503 is Clark Kent's Metropolis apartment number in the "Superman" series.

Source: Prime Curios!

127

127 is a prime number.

127 is a Mersenne number: 2⁷ – 1 = 127. It is the largest Mersenne prime with all different digits.

127 = 2⁰ + 2¹ + 2² + 2³ + 2⁴ + 2⁵ + 2⁶.

127 can be expressed as the sum of factorials of the first three odd numbers: 1! + 3! + 5! = 127.


127 is the number of prime days in a leap year.

Source: Prime Curios!

12,837,064

12,837,064 = 2³ x 631 x 2543 (via WolframAlpha)

12,837,064 is the number of decimal digits in the 47th known Mersenne prime, discovered April 12, 2009.

This prime is at present the second largest known prime number, a mere 141,125 digits smaller than the record-breaking Mersenne prime found last August.

Source: Great Internet Mersenne Prime Search

178

178 = 2 x 89. Note that 2 and 89 are the smallest and the largest Mersenne prime exponents under 100.

178 has a cube with the same digits as another cube: 178³ = 5,639,752 and 196³ = 7,529,536.

Each internal angle of a regular 180-sided polygon is 178 degrees.

178 has a representation as a sum of two squares: 178 = 3² + 13².

The Heinkel He 178 was the world's first aircraft to fly under turbojet power, and the first practical jet plane.

Source: Numeropedia

31

31 is a prime number. It is also a Mersenne number (of the form 2^p - 1, where p = 5).

31 is the sum of the first two primes raised to themselves: 31 = 2² + 3³.

31 is the smallest number with exactly two representations as a sum of four squares: 31 = 1² + 1² + 2² + 5² = 2² + 3² + 3² + 3².

There are only 31 numbers that cannot be expressed as the sum of distinct squares.

29 and 31 form a twin prime pair.

31 is 11111 in base 2 (binary).

31 = 5⁰ + 5¹ + 5², which means it is 111 in base 5.

31 is a number that cannot be written as a sum of three squares.

31 is the minimum number of moves required to solve a Tower of Hanoi puzzle containing five discs. The Tower of Hanoi puzzle consists of three rods and a given number of discs that can slide onto any rod. The discs start as an ordered stack on one rod, with the largest at the bottom and the smallest on top. The objective is to move all the discs from one rod to another rod, with the discs again stacked in the correct order, subject to the following rules: only one disc may be moved at a time; each move consists of taking the upper disk from one of the rods and sliding it onto another rod, on top of the other disks that may already be present on that rod; and no disc may be placed atop a smaller disc.


Source: Number Gossip; Cut the Knot

11 185 272

11 185 272 is the number of decimal digits in the 46th known Mersenne prime, discovered on Sept. 6, 2008, by Hans-Michael Elvenich in Germany, a participant in the Great Internet Mersenne Prime Search.

A Mersenne prime has the form 2^p – 1, where p is a prime number. The 46th Mersenne prime, the second largest identified to date, is 2^37156667 – 1. A larger Mersenne prime, 2^43112609 – 1, was discovered on Aug. 23, 2008. The 46th Mersenne prime is the first prime to be discovered out of order since Walt Colquitt and Luther Welsh Jr. found 2¹¹⁰⁵⁰³ – 1 (the 29th Mersenne prime) in 1988.

Source: Great Internet Mersenne Prime Search; Mersenne Primes

12 978 189

12978189 = 3² x 7² x 29429.

12978189 has a representation as a sum of two squares: 12978189 = 483² + 3570².

12 978 189 is the number of decimal digits in the world’s largest known prime number, discovered on a UCLA computer on Aug. 23, 2008, as part of a 12-year-old, worldwide computing project called the Great Internet Mersenne Prime Search (GIMPS).

This number is the 45th known Mersenne prime, 2^43112609 – 1.

Marin Mersenne (1588–1648)

Source: Great Internet Mersenne Prime Search.
Image: Courtesy of Loci: Convergence