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# 11

The only palindromic prime with an even number of digits due to the fact that all palindromes with an even number of digits are divisible by 11.

(11^{11} * 2^{11} + 1)/(2^{11} - 1) is prime. [Luhn]

There's a curious way to multiply a two digit number by the prime number 11. Add the two original digits and place this sum between them. E.g., 35 * 11. (3 + 5 = 8) and we get 385. Note that if the sum of the two digits exceeds 9 place the units digit of this sum between the two numbers, and add 1 to the hundreds digit. [Franco]

The smallest number with multiplicative persistence of 11 is 277777788888899. [Sloane]

The secret formula for Kentucky Fried Chicken includes 11 herbs and spices. [Sanders]

Sunspot activities seem to follow an 11-year cycle.

11^{11} contains exactly two embedded elevens.

UPC bar codes consist of 11 digits (not counting the "check digit"). [Bobick]

The 11th prime times π(11) has a digit sum of 11.

Half of the first sixty-four partition numbers are divisible by 11. [Wilson]

Rotakas, spoken in the center of Bougainville Island in the South Pacific, uses only 11 phonemes.

The smallest anti-Yarborough prime, i.e., a prime containing only the digits 0 and 1.

11 has the primitive root *g* = 2, meaning the remainders after dividing 2^{n} by 11 assume all values 1, 2, ... 10, exactly once as *n* goes from 1 to 10.

Divisibility test for 11: Combine the digits in order by alternating them with positive and negative signs. If the result is divisible by 11, then so is the original number. For example, 11 divides 90816 because 11 divides + 9 - 0 + 8 - 1 + 6. [Beedle]

Five consecutive powers of 11 produce palindromes:

11^{0}= 1 11^{1}= 11 11^{2}= 121 11^{3}= 1331 11^{4}= 14641.

"Elevenses" is a British and colonial meal that is similar to afternoon tea but eaten around 11 o'clock in the morning.

The number of cards including the Significator (the focus card) typically used in a Tarot reading. [Haga]

The first repunit prime. The term repunit (coined by A. H. Beiler in 1966) comes from the words *rep*eated and *unit*, so repunits are positive integers in which every digit is 1.

The smallest two-digit additive prime (a prime whose sum of digits is prime). [Russo]

There are 11 stars in Vincent van Gogh's *Starry Night*. Was the great painter inspired by Genesis 37:9? [Easterling]

11 + 2*n*^{2} gives primes over the first 11 whole numbers.

11 divides its previous consecutive concatenated composite numbers (8910). [De Geest]

Paul Erdős observed that both 3*4 and 5*6*7 are congruent to one modulo 11.

The repunit 11111111111 (11 ones) has only two prime factors.

If *n* is sufficiently large, then between *n* and *n* + sqrt(*n*) there exists a number with at most 11 prime factors. [Brun]

Aibohphobia (the fear of palindromes) is palindromic itself and contains 11 letters. [Patterson]

At least one edge of the hypothetical perfect cuboid (or Euler integer brick) is divisible by 11. [Beedassy]

The number of letters in PRIME NUMBER. [Kumar]

11 is the (1+1)nd odd prime that cannot be expressed as the sum of two prime numbers. [Astle]

10! - 11 and 10! + 11 are two consecutive primes. [Firoozbakht]

The only number in base ten that is not Nivenmorphic ; In other words, 11 never divides a number ending in 11 if its sum of digits is also equal to 11. [Boscaro and Beedassy]

The largest example where p divides (p - 1)# - 1. [Luhn]

Today only 11 lines of Sotades the Obscene of Maronea's works still remain. Most sources credit him with inventing palindromes in Greek-ruled Egypt, back in the 3rd century B.C.

The minimum prime p such that p ± 1 each has exactly 2 distinct prime factors. [Das]

The number 11 London Bus runs from Fulham Broadway to Liverpool Street Station. [Croll]

The Klein cubic threefold is the only smooth cubic threefold which has an automorphism of order 11. [Post]

The only prime p such that p ± 8 and p ± 6 are all prime. [Murthy]

A letter to the editor of "The Times" newspaper on the partially palindromic date of 22 February 2002 contained the 11 character palindrome "Madam I'm Adam" (excluding spaces and punctuation). [Croll]

The smallest prime with persistence (multiplicative as well as additive) of 1. [Gupta]

The smallest prime (the fifth) which is greater than the corresponding (the fifth) composite, which is 10. [Murthy]

The smallest prime for which the sum of digits equals the number of digits. [Murthy]

The smallest prime *p* such that 2^{p} - 1 is composite. [Russo]

11 is the smallest prime of the form n ns followed by 1. Note that eleven 11s followed by 1 is also prime. [Poo Sung]

The smallest odd prime which can be represented as sum of a triangular number and its reverse, i.e., 10 + 01 = 11. [Gupta]

11 is the smallest prime that is equal to the product plus sum of two primes (2*3 + (2+3) = 11). [Wagler]

The largest known example such that p# ± 1 are both primes. p# is primorial p. [Gupta]

The smallest multidigit Lucasian prime: p is congruent to 3 (mod 4) with 2p+1 prime. [Russo]

The smallest prime p such that 6*p-1 is composite. [Russo]

The smallest palindromic prime p such that 2^p-1 has at most 2 prime factors. [Russo]

The smallest odd prime which can be represented as the sum of a number and its reverse, i.e., 10 + 01 = 11. [Gupta]

The smallest strong prime. A strong prime is a prime that is greater than the arithmetic mean of the nearest prime above and below. [Russo]

11 = 3! + 5. Note that 3 and 5 are the first twin primes. [Avrutin]

11 is the smallest prime p such that sigma(x) = 2^p has no solution. [Gupta]

11=1+2+8 and the only known values of n for which (p(n)^p(n)-1)/(p(n)-1) is prime are n=1,2,8, and 11. [Rupinski]

The largest number (coincidentally prime) which is not expressible as sum of two composite numbers. [Gupta]

The smallest prime p such that 6p ± 1 do not form a twin prime pair. [Rupinski]

The smallest prime *p* such that the sum of digits of
*p* divides the sum of digits of the *p*th prime. [Beedassy]

11^2 = 37 + 41 + 43 (i.e., the sum of three consecutive primes). [La Haye]

2^^n-9 = 2^(2^(2^...(2^2)..))-9 is divisible by 11 for all sufficiently large n. 11 is the largest prime less than a million of which this is true. [Hartley]

11 is the earliest prime p such that p^3 is the sum of 3 consecutive primes: 11^3 = 439 + 443 + 449. [Rivera]

ElevenSmooth is an online distributed computing project searching for prime factors of *M*(3326400).

In "Western" music's 12-tone scale, ratios of fractions between 2 and 1 with a one-digit denominator that involve 11 are not used. The ratios 11/6, 11/7, 11/8, and 11/9 are not used in "Western" music. Neither are fractions with 11 as the denominator. These notes are not found in the "Western" twelve-tone scale, but might be found in the music of other cultures that use different tuning systems. [Obeidin]

The original formulation of M-theory was in terms of a (relatively) low-energy effective field theory, called 11-dimensional supergravity.

Self-proclaimed psychic Uri Geller (1946- ), has spoken repeatedly about the occurrence of two 11's side-by-side. For example, the bizarre attraction some people have for the time 11:11 on digital clocks and watches.

N = 11 is the largest prime with the following property: N# is written with n-1 distinct digits, where N is the nth prime. [Necula]

The doubly-true solution to the following English alphametic, a type of cryptarithm in which the letters make sensible words or phrases: THREE + THREE + TWO + TWO + ONE = ELEVEN. The object is to replace the letters of the alphabet with decimal digits to make a valid arithmetic sum. The term "alphametic" was coined by J. A. H. Hunter in 1955.

11^11+11^11*11-1 is prime. Note that 11 ones are used in the previous expression. [Patterson]

The smallest prime formed from two Fibonacci numbers. [McAlee]

If T(A,B) = (A-B^2)^2-B with B < A, then 11 is the smallest value of A such that [A, A+2], [B, B+2], and [T(A,B), T(A,B)+2] are all twin prime pairs (in this case, B=5). All twin primes will exhibit this property if it is allowed that B > A. [Opao]

There are eleven two-digit primes ending in a prime digit. [Silva]

A hendecasyllabic is verse written in lines of exactly 11 syllables. [Patterson]

11 is the smallest prime equal to the product of twin primes minus their arithmetic mean (3*5 - 4 = 11). [Teofilatto]

11 is the smallest prime of the form 2^p + p, where p is prime. [Teofilatto]

11 is the only prime of the form 3^p + p for p a prime. [Patterson]

There are exactly 11 positive odd integers N that cannot be expressed as the sum of two coprime composite integers, both of which are less than N. [Rupinski]

The largest prime number that cannot be written as the sum of four hexagonal numbers. Note that it is the only prime number that cannot be represented using five hexagonal numbers. [Capelle]

11 is the smallest number of sides of a regular polygon that cannot be packed about a point with other regular polygons. E.g., a 7-sided polygon packs with a 42-sided polygon and a triangle. [Astle]

*The Works of Charles Babbage*, published in London by Pickering and Chatto Publishers, is an 11-volume set. [McAlee]

The sum of the 11 primes after 11 is another palindromic prime. [Silva]

World War I ended formally at 11 A.M. on the 11th day of the 11th month of the year. Now the date is celebrated as Veteran's Day. [McCranie]

There is no number of the form n^2+n+1 that can be evenly divided by 11. [Schuler]

11 Downing Street in London is the official residence of the Chancellor of the Exchequer. [McCranie]

The badge number of Napoleon Solo in the TV show "The Man From U.N.C.L.E." is 11. [McCranie]

The name for the now dwarf planet Pluto (discovered by Clyde W. Tombaugh in 1930) was proposed by 11-year-old Venetia Burney of Oxford, England. She is now a retired teacher whose married name is Venetia Phair. [Paddy]

Joseph-Louis Lagrange (1736--1813) is widely regarded as the finest mathematician of the 18th century. He was the first-born of 11 children.

The largest integer that cannot be expressed as a sum of (two or more) distinct primes. [Capelle]

United States Mint image |

The 2002 All-star baseball game was declared a tie after 11 innings (a rare and controversial occurrence). [McCranie]

The sum of first 11 primes ending with 1 is a prime that begins with 11. [Bopardikar]

Substance P is an 11-amino acid polypeptide that has been associated with the regulation of stress brought about by failure to find large primes.

The smallest tetradic prime. [Capelle]

The smallest odd Ramanujan prime. [Beedassy]

The first number *n* that divides N-1, where N is the least common multiple of all numbers below *n*. [Beedassy]

The rock group Spinal Tap has amplifiers that "go to 11." [McCranie]

Fasching (Carnival) begins in the Rhineland of Germany on the Eleventh minute of the eleventh hour of the eleventh day of the eleventh month (In German: Am elfte elfte elfte elfte!) [Fellows]

There are 11 schools in the "Big 10" Conference. [McCranie]

There is no known number with multiplicative persistence > 11. [Capelle]

Golomb proved that *n*/π(*n*) takes on all
integer values *k* > 1 at least once. Currently, *k*=11 is the only number known for which there is exactly one solution. [Caldwell]

A pack of Live Savers candy contains 11 pieces. [McCranie]

11!+11!+11!+11!+11!+1 is prime. One is repeated 11 times in the expression. [Silva]

Ocean's "11" is a film with George Clooney and Brad Pitt. This film won 3 awards and it had 17 nominations. (3 and 17 are prime also.) [Montardit]

The first prime which cannot be a pandigital fraction (including or excluding 0). [Poo Sung]

The smallest number that is not an idoneal number.

The smallest counterexample to Mersenne's conjecture is the Mersenne number M(11) = 2^11 - 1 = 2047 = 23 * 89, which is a semiprime (and brilliant) rather than a prime, even though 11 is a prime. [Post]

11 is the only prime of the form p*q+r, where p, q, r are consecutive primes. [Silva]

Any number of the form ABC...ZZ...CBA is divisible by 11. [Neves]

The only prime that can be expressed by two consecutive primes in the forms p^q+q and q^p+p. [Silva]

11^11 - π(11)! is prime. It is the largest known prime with this property. [Javadifar]

11 ones minus 11! is an 11-digit prime with 11 as the first and last digits. [Silva]

11 = π(11) + π(π(11)) + .... It is the largest such number. [Firoozbakht]

The only known multidigit prime of the form prime(n)^n + n. The next such prime if it exists has more than 30000 digits. [Firoozbakht]

11^(ln(11)) is approximately 100*π. 11^(ln(11)) = 314.159789 and 100*π = 314.159265 (6-digit accuracy). [Wehner]

The smallest prime which when sandwiched between a two-digit repdigit gives a multiple of 11. In other words 1111, 2112, 3113, 4114, 5115, 6116, 7117, 8118, and 9119 are multiples of 11. [Meller]

The smallest number of a finite set of numbers inscribed on the faces of a Mughal gold box in the shape of an icosahedron: 11, 20, 21, 31, 41, 51, 61, 71, 81, 91, 101, 201, 202, 301, 401, 501, 601, 701, 801, and 901. An accompanying manuscript note records that the box was found in Tipu Sultan's treasury, after the fall of Seringapatam (India) in 1799. Some research has been devoted to the interpretation of the 20 Arabic numerals of this box, which at first appeared to be unrelated. Among the 11 composite numbers of this list, what's the smallest? [Capelle]

The largest prime whose cube (1331) can be displayed on a 24-hour digital clock. [Fellows]

The 11-cell (or hendecachoron) is a self-dual abstract regular 4-polytope (four-dimensional polytope).

Enter any 11-digit prime number to continue...

The smallest prime *p* whose next *p* primes
starting with *p* add up to a prime (emirp) ending in
*p*: 11 + 13 + ... + 43 + 47 = 311. [Beedassy]

The smallest prime whose digits are both nonprime and noncomposite. [Beedassy]

ELEVEN is the only prime word containing a smoothly undulating vowel/consonant pattern that begins with a vowel. [Cash]

Internet users who visited Google's website on September
27, 2009, saw the Google logo spelled "Goog**ll**e" in
celebration of the search engine giant's 11th birthday.

The smallest number of divisors which no number can have among the first thousand. [Beedassy]

The first 1+1 digits of π(11^11) are 11. [Firoozbakht]

The first prime of the smallest pair of non-trivial reversible twin primes. [Silva]

There are eleven five-digit palindromic primes formed from prime digits. [Silva]

The highest prime dividing the number of yards, feet, or inches (basic US length units) that make up a mile. [Beedassy]

The Apollo 11 mission put the first Earth man on the Moon. [Green]

11^n, for natural numbers n, will produce the binomial coefficients: 11^1 = 11, 11^2 = 121, 11^3 = 1331, etc. [Giberson]

2! + 3! + 5! = 7 (mod 11). There exists only one other known prime with this property. [Firoozbakht]

Some Russian leaders believe that 11 time zones across their country is too many. [Green]

The smallest straight-line prime. [Pol]

2*3*5*...*prime(11) = 11 (mod 11+1). It is the only known prime with this property. [Firoozbakht]

The only known palindromic Wagstaff prime. Note that it creates as exponent another Wagstaff prime. [Loungrides]

There are 11 major lakes in the Finger Lakes Region of Upstate New York. [Teague]

The kitchen at Prime Restaurant in Huntington, Long Island, NY, remains open until 11 PM on weekdays.

Probably the only prime whose cube is formed from the next prime after it and its reversal. [Silva]

12345678910 minus 11 is an eleven-digit prime. [Silva]

The maximum number of regions formed inside a pentagon when cut by its diagonals. [Green]

π(π(1) + π(2) + ... + π(11)) = 11. It is the only number with this property. [Firoozbakht]

The smallest multidigit Zuckerman number that is prime. (A Zuckerman number is a number n which is divisible by the product of its digits). [Loungrides]

11 = 1_{2} + 11_{2} + 111_{2}. [Poo Sung]

There are 11 capital letters in the English alphabet that have vertical symmetry. Can you find them all? [Cook]

Obtained by subtracting the sum of all digital primes from the sum of all non-prime digits. [Silva]

The largest of only two non-titanic primes of form p^(p-1)+(p-1), for p a prime (case p=3). The other is 3. [Loungrides]

There is only one case of three consecutive primes all ending in 11 among the first billion primes. Can you find them? [Gaydos]

The smallest odd prime whose binary expansion is not palindromic. [Pol]

The RMS *Titanic* had 11 decks.

The smallest prime whose factorial ends in prime number of zeros. [Gupta]

The number of nonsymmetrical fundamental solutions to the eight queens puzzle.

The largest prime number with a single-morpheme name.

The largest prime of form 1!+2!+ … +n!+2, (n=3). Note that every other number of this form, for n > 3, ends with 5 [Loungrides]

The only prime that cannot be represented using five hexagonal numbers. [Deza]

2^1+3^2 is the only non-titanic prime of form p^(p-1)+q^(q-1)+r^(r-1)+s^(s-1)+..., where (p, q, r, s, ...) is the sequence of the first consecutive primes. [Loungrides]

Ivan M. Niven (of Niven numbers fame) was selected to write the Carus Monograph Number 11, entitled *Irrational Numbers*.

Checkmate by promotion of Pawn to Knight can occur in 11 half-moves in unique sequence excepting a possible 'en passant' capture. [Merickel]

The *Puzzle Prime* website has an 11x11 Grid brain teaser.

There are eleven strobogrammatic numbers with distinct digits (0,1,8,69,96,609,619,689,906,916,986). [Gaydos]

Take all twin primes (including the "Siamese twins" 2,3) less than 11, namely: (2,3), (3,5), (5,7). If we square each prime and then add the squared prime pairs together, we get 121. Or, 11^2. [Missailidis]

Sir Frederick Pollock conjectured that every natural number is the sum of at most 11 centered nonagonal numbers. This is almost certainly true, but has not yet been proven. [Homewood]

The *YouTube* online unlimited video-sharing platform
uses a 11-character randomly generated code in base 64 to
identify each upload. [Beedassy]

11 is the 5th prime. Five is the 3rd prime. Note that 3 is the 2nd prime. [Roberto]

Joseph Smith obtained testimonies from 11 men who said that they had seen the golden plates. [Tedrick]

The smallest strobogrammatic prime. Can you find a prime formed from the concatenation of the first n strobogrammatic primes?

The custom essay writing service "**Prime**-Writing.com."
provides an urgency level of up to 11 days.

Nearly half of the length of the book "Prime Recreations: An Olio of Curios about Prime Numbers" by Bruce Pyne, is devoted to 11 chapters of lists of primes.

There are 11 prime quintuplets of the form {p, p + 4, p + 6, p + 10, p + 12} less than 10^5. [Homewood]

There are eleven distinct nets on a plane that can be folded up to produce a cube. [Beedassy]

Great Britain lost 11 days in 1752 because of the Julian to Gregorian Calendar change. [Gupta]

The PDP-11 is considered by some experts to be the most popular minicomputer ever. [Allen]

The maximum number of turns required to solve the Mini Cube (the 2×2×2 equivalent of a Rubik's Cube) is up to 11 half or quarter turns. [Beedassy]

The only Sophie Germain prime of the form 4*p - 1, where p is prime. [Xavier]