13

13 is a prime number.

13 has a representation as a sum of two squares: 13 = 2² + 3².

13 is the 7th Fibonacci number.

11 and 13 form a twin prime pair.

13 is a lucky number. To build the lucky number sequence, start with natural numbers. Delete every second number, leaving 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, . . . . The second number remaining is 3, so delete every third number, leaving 1, 3, 7, 9, 13, 15, 19, 21, . . . .The next number remaining is 7, so delete every 7th number, leaving 1, 3, 7, 9, 13, 15, 21, . . . . The next number remaining is 9, so delete every ninth number, and so on. The numbers that remain are lucky: 3, 7, 9, 13, 15, 21, 25, . . . .

13² = 169 and (reversing all digits) 31² = 961.

Three planes can cut a doughnut into a maximum of 13 parts.

13 is 111 in base 3.

The fear of the number 13 is called triskaidekaphobia.

There are 13 tricks in each hand of bridge.

Source: Number Gossip; Prime Curios!

13131

13131 = 3² x 1459.

13131 is an octahedral number.

13131 is palindromic in base 10 and in base 8 (31513).

13131 is a palindrome n such that 6n + 1 is also a palindrome: 78787.

13131 is the ninth-largest palindrome whose digit sum is 9.

13131 is a nonprime such that the sum of its digits equals the product of its digits: 1 + 3 + 1 + 3 + 1 = 9 = 1 x 3 x 1 x 3 x 1.

13131 is the zip code for Parish, N.Y.

Source: On-Line Encyclopedia of Integer Sequences

1313

1313 = 13 x 101.

1313 is an evil number because it has an even number of 1s in its binary expansion: 10100100001.

1313 has two representations as a sum of two squares: 1313 = 17² + 32² = 23² + 28².

1313 is the hypotenuse of two primitive Pythagorean triples: 1313² = 255² + 1288² = 735² + 1088².

1313 is a divisor of 100⁶ - 1.

The starburst galaxy NGC 1313 is known as the Topsy-Turvy Galaxy.


The Italian poet Giovanni Boaccaccio was born in the year 1313.

Source: ESO
Image: (c) ESO

313

313 is a prime number, as is 10³¹³ + 313.

313 has a representation as a sum of two squares: 313 = 12² + 13².

If 313 people are chosen at random, then the probability that at least five of them will share the same birthday is greater than 50%.

313 is 100111001 in base 2 (binary).

311 and 313 form a twin prime pair.

313 is the hypotenuse of a primitive Pythagorean triple: 313² = 25² + 312².

In cartoons featuring Donald Duck, his car often sports the license plate number 313.

Source: Prime Curios!

1969

1969 = 11 x 179.

1969 is a composite number not ending in zero that yields a prime number when turned upside down: 6961 is prime.

1969 is the only known counterexample to a conjecture about modular Ackermann functions.

Now useful in computer science, the Ackermann function was first studied by David Hilbert and Wilhelm Ackermann about 90 years ago. It grows faster than an exponential function or even a multiple exponential function.

The Ackermann function A(x, y) is defined for integer x and y in the following way:
A(x, y) is y + 1 if x = 0; A(x – 1, 1) if y = 0; A(x – 1), A(x, y – 1) otherwise.

We can also restrict the range of the Ackermann function to a finite set, defining the mod-n Ackermann function. For a given n, the numbers initially bounce around wildly as x increases, but nearly always stabilize to a certain value or an alternating pair of values.

According to Jerrold W. Grossman of Oakland University, the only value of n less than 4,000,000 for which the standard mod-n Ackermann function does not stabilize is n = 1969. Why this is so remains a mystery.

Neil Armstrong became the first man to step onto the moon's surface on July 20, 1969.

Source: Jon Froemke and Jerrold W. Grossman. 1993. A mod-n Ackermann function, or what’s so special about 1969? American Mathematical Monthly 100(February):180-183.

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

1221

1221 = 3 x 11 x 37.

1221 is a number that can be expressed as the product of numbers made of only one digit: 1221 = 11 x 111.

1221 is a divisor of 100³ - 1.

1221 is a number of the form n² - 4, for n = 35.

1221 is 14341 (palindromic) in base 5. 1221 is 5353 in base 6, 3363 in base 7, and 633 in base 14,

In the palindromic year 1221, the Maya of the Yucatán revolted against the rulers of Chichen Itza and Holy Roman Emperor Frederick II founded the University of Padua in Italy.

Source: Wikipedia