# 313

This number is a prime. Ever since Donald Duck's car first appeared in 1938 (a "1934 Belchfire Runabout" that he built from spare parts), it has frequently sported the license plate number 313. [Haga] 313 (base ten) = 100111001 (base two). Note that 100111001 (base ten) is a palindromic prime as well. The only three-digit number with this property. [Larsen] The largest known prime that divides a unitary perfect number. The telephone area code serving the city of Detroit, Michigan. It was popularized in the hip hop drama film 8 Mile starring Eminem. [Anon] The smallest multidigit palindromic prime of the form n * phi(n) + 1, where phi(n) is the Euler function. [Russo] Islam's first battle (The Battle of Badr) against the pagans of Mecca was fought and won by 313 Muslims. In Shi'ism, it is believed that Imam al-Mahdi (the ultimate savior of the believers) will appear when there are 313 true and sincere Shia followers in the world. [Daniyal] Frenicle challenged Wallis to solve x2 - 313y2 = 1. [Beiler] Elizabeth Haich's book Initiation analyzes the significance of 313 in ancient Egyptology. 313 = 122 + 132. [De Geest] 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%. Wood's History of Alameda County, California, published in the decade around the first U.S. centennial, includes 313 men and not one woman! [Haga] The smallest weakly prime in base 5. [Duisenberg and Dionne] 313 * 2519748 + 1 is prime. The smallest number to appear exactly three times in its own factorial (313!). [ten Voorde] 10313 + 313 is prime. [Broadhurst] The Mayan ruins of Cobá near Tulúm in the Yucatán peninsula contains a pyramid that has a total of 313 glyphs on its four sides. [Haid] In the movie Somewhere In Time, playwright Richard Collier (Christopher Reeve) checks into room number 313 of the Grand Hotel (Mackinac Island, Michigan). [Haga] The smallest of three 3-digit palindromic primes that are not the sum of consecutive composite numbers. The other two are 383 and 757. [Russo] (2^3+1)/3, (2^31+1)/3, (2^13+1)/3, and (2^313+1)/3 are prime. [Aggarwal] The smallest happy number which is a multidigit palindromic prime. [Gupta] 1/313 produces a string of decimals that repeats itself after 312 (one less than itself) decimals. Only 5 of the 3-digit palindromic primes have this ability. [Vatshelle] 313, 313 - 2, 313 - 2*3, 313 - 2*3*5 & 313 - 2*3*5*7 are all prime numbers. [Rivera] The smallest multidigit palindromic prime that is both a left and right-truncatable prime. [Patterson] The paper Light and Number: Ordering Principles in the World of an Autistic Child by Park and Youderian begins on page 313 in the Journal of Autism and Childhood Schizophrenia (1974). Note that the Park's child Elly was fascinated by the "order" of numbers, especially primes. The Roman emperor Constantine made Christianity the official religion in A.D. 313. [Pitts] There are 313 exclamation marks in the KJV Bible! The first one occurs at the end of Genesis 17:18. [Bennet] 313 is the 13th centered square number. Note that 313 left-truncates to 13, which is also a prime. [Post] 313 "inverted" is also a prime, 131. [Blackham] The smallest positive integer solution to 313(x3 + y3) = z3 contains numbers that are titanic. Eris, the largest known dwarf planet in the solar system, was previously designated UB313 in 2003. 313 is the name of a unique wakeboarding trick: "Heelside Raley with a frontside handle-pass 360." [Hill] The sum of the largest Fibonacci square (144) and the largest Pell square (169). [Gupta] 313 -> 3 * 13 = 3 + 5 + 7 + 11 + 13. [Vrba] 313 remains prime if its digits are replaced by their squares, cubes or fourth powers. [Silva] A fingerprint of Divinity is seen in this number in that it is a compound of two mirror-image emirps, 31, the Hebrew alphanumeric value of the name GOD [EL = 1+30], and 13, the prime in the Hebrew name of Jehovah [YHVH = 10+5+6+5 = 26 = 2*13], hence, 31/13 compounded = 313. [Pitts] The smallest palindromic prime number p such that reversal(p^2) is prime. [Firoozbakht] 313 divides 102345679. [Wesolowski] The seventh prime whose sum of digits is 7. [Wesolowski] Palindromic prime formed from the reflected decimal expansion of π. [Silva] Subject of the psychedelic artwork featured in the 2010 October Edition of Pour la Science. The magic sum of the smallest possible order-5 prime simple magic square. The smallest half-quartan palindromic prime, i.e., (1^4+5^4)/2 = 313. [Loungrides] The number of intersections formed when the diagonals of a regular dodecagon are drawn. [Gudipati] The only non-titanic prime formed by concatenating in ascending order the first n consecutive numbers ending with 3 (n=2). [Loungrides] The smallest prime with 10 as least positive primitive root. [Wesolowski] The view from Dead Horse Point (off of State Route 313 in Utah) is one of the most photographed scenic vistas in the world. Smallest multidigit palindromic prime whose English name contain prime number of letters (i.e., "Three hundred and thirteen" contain 23 letters). Also 23 is smallest multidigit prime whose English name contain palindromic prime number of letters. [Gupta] The smallest prime such that the reversal of its square (96979) is prime. Note that rev(96979^2) is prime also. [Loungrides] Pi Day Eve is 3/13. [Beedassy] Zapruder sold the print rights of his famous film to Life Magazine with the condition that frame 313 be removed. 313^313+(31+3)^85 is prime. Note that 8+5=13. [Sariyar] The month and day are NOT simultaneously prime a total of 313 times in both leap years and non-leap years [Homewood] The first occurrence of 313 in π happens at position 137 (the multiplicative inverse of the Fine Structure Constant). [Gooch] The smallest prime number that is a substring of the concatenation, in either order, of the next two primes (331 and 317). The next four such primes are 111091, 777781, 1131113, and 3333313. [Gaydos]