# 17

This number is a prime.

Prime Numbers CD Image J. Neptune © 1994 |

*jūshichi-gen*, literally "seventeen strings") is used in the album

*PRIME NUMBERS*by Neptune/Watanabe. As a matter of fact, all the traditional Japanese instruments played in it contain prime numbers. The other instruments used are the 3-stringed shamisen, the 5-holed shakuhachi (bamboo flute), and the 13-stringed koto.

The only known prime that is equal to the sum of digits of its cube (17^{3} = 4913 and 4 + 9 + 1 + 3 = 17). [Gupta]

Primonumerophobia (the fear of prime numbers) contains 17 letters.

The only prime that is the average of two consecutive Fibonacci numbers. [Honaker]

In 1796, Gauss discovered the construction of a regular 17-gon with compass and straightedge.

"Seventeen" was the original title of The Beatles' song "I Saw Her Standing There." It was written on a Liverpool Institute exercise book. [McCranie]

The smallest odd number greater than three that cannot be represented as the sum of a prime and twice a nonzero square. [Stern]

"17-jewel watches" have hard gems at 17 bearing points of friction.

Frank Bunker Gilbreth (1868-1924), pioneer of modern motion study technique, and his wife Lillian devised a classification scheme to label 17 fundamental hand motions of a worker, which they called *therbligs* ("Gilbreth" spelled backwards with the *th* transposed).

Theodorus of Cyrene (5th century B.C.) proved that the square root of each of the odd primes up to 17 is irrational. It is not known why he stopped at 17.

Pluto's orbit around the sun is inclined approximately 17° to the plane of the solar system. [Hartmann]

(17 * 2^{9251})^{2} + 1 is a prime of shape *a*^{2} + 1. [Keller]

17 is the smallest natural number that is written in French as a compound word: *dix-sept*. [Lefèvre]

The least integer such that the sum of its digits in every base B = 2, 3, 4, 5, 6, 7, 8 is prime. [Crespi de Valldaura]

In 1951 Ferrier found that (2^{148} + 1)/17 is prime.

The smallest prime that is both the sum of a prime number of consecutive composites, and also, the sum of a composite number of consecutive primes: 17 = 8 + 9 = 2 + 3 + 5 + 7. [Beedassy]

The least prime that is the sum of two fourth powers, 1^{4} + 2^{4}. [Crespi de Valldaura]

The world's largest caldera is that of Mt. Aso in Kyushu, Japan, which measures 17 miles north to south and 71 miles in circumference. (17 and 71 are emirps.)

The 17-year locust has the longest cycle of development of any known insect. Note that its Latin name (*Cicada septemdecim*) has 17 letters. The genus *Magicicada* is sometimes called the "seventeen-year locust,"
but they are not locusts; locusts belong to the order Orthoptera. [Dillon]

The smallest prime of the form *p*^{2} + *q*^{3}, where *p* and *q* are primes. [Russo]

17 * 65359477124183 = 1111111111111111.

In any set of *k* < 17 consecutive integers there is always one
which is relatively prime to all the others. [Pillai]

The Parthenon is 17 columns long.

17^{2} can be expressed as the sum of 1, 2, 3, 4, 5, 6, 7, 8 distinct squares. [Crespi de Valldaura]

There are 17 plane symmetry groups, i.e., there are 17 different ways that a wallpaper design can repeat.

2^{n} - 17 is prime for *n* = 6, 8, 22, 44, 414, 444, 808 and 848, not to mention the non-palindromic values of *n*. [Kulsha]

The smallest two-digit prime of the form F(*i*)^{3} + F(*j*)^{4}, where F(*n*) are
Fibonacci numbers. [Russo]

17 is the *only* prime of the form *p*^{q} + *q*^{p}, where *p* and *q* are prime. [Kulsha]

17 + 2^{1}, 17 + 2^{1} + 2^{2}, 17 + 2^{1} + 2^{2} + 2^{3}, 17 + 2^{1} + 2^{2} + 2^{3} + 2^{4} and 17 + 2^{1} + 2^{2} + 2^{3} + 2^{4} + 2^{5} are primes. [Luhn]

Stegosaurus had 17 bony plates that were embedded in its back.

The only number *n* having *n* partitions into prime parts. [Beedassy]

Friday 17th is unlucky in Italy. This is thought to derive from Roman tomb stones which often included the word VIXI (I lived) as part of the narrative (VI + XI = 17). [Bassano]

Using the alphabet code: A = 1, B = 2, ... , Z = 26, 17 is the smallest nonnegative number whose numerical value of its word form is also a prime (109). [Trotter]

17 = 3^{4} - 4^{3}.
[Murthy]

17 is the middle term of the longest arithmetic progression of primes with a common difference of 6. [Murthy]

Moderately active people can estimate their daily calorie requirement by multiplying their weight in pounds by 17. [Pierson]

President Bill Clinton's dog Buddy was killed by a vehicle driven by a 17-year-old girl.

"At Seventeen" was a hit song for Janis Ian in 1975. [Litman]

"The prime number 17 accommodates all the positive-negative, quanta-wave primes up to and including the number 18, which in turn accommodates the two nines of the invisible twoness of all systems." (Quoted from from Section 1238.43 of Synergetics by R. Buckminster Fuller.) [Croll]

The smallest prime whose sum of the digits is a cube. [Murthy]

Marin Mersenne's vast correspondence (*Correspondance du Père Marin Mersenne, religieux minime*) fills a total of 17 volumes. [Caldwell]

The first four digits of 2^{(17+17)} is 1717. [Wu]

(17^17+1)/(17+1) is prime. [Poo Sung]

There are 4 ways to express 17 as p+2q; p, q primes. [Russo]

The smallest prime formed from concatenation of happy numbers. [Gupta]

(2^17-1)/(2-1) and (2^17+1)/(2+1) are both primes. [Dobb]

No odd Fibonacci number is divisible by 17. [Honsberger]

17 is the smallest prime sandwiched between two non-squarefree numbers. A number is *squarefree* if it is not divisible by the square of an integer greater than 1. [Gupta]

17 and apple: There are 17 sets of chromosomes in an apple. Eve was traditionally 17 years of age when she fed Adam the apple. Note that there were 17 different PowerMacs made by Apple Computer. [Faulds]

17 is the only odd prime p such that p + sod(p) and p - sod(p) are square numbers (5^{2} and 3^{2}, respectively). [Rivera]

17 is the smallest prime for which the decimal expansion of the reciprocal 1/17 contains all 10 digits. [Hartley]

Schnizel showed that Goldbach's conjecture is equivalent to saying that every integer greater than 17 is the sum of three distinct primes.

There are exactly 17 ways to express 17 as the sum of one or more primes. [Rupinski]

"Seventeen or Bust" is a distributed attack on the Sierpinski problem. [Mendes]

Gauss showed that constructing a 17-sided polygon is tantamount to solving the equation *x*^{16} + *x*^{15} + ... + *x* + 1 = 0. [Stewart]

In the Aardman film *Chicken Run*, the chickens did much of their plotting in Hut 17. [Hartley]

17 steps lead up to Sherlock Holmes' residence at 221B Baker Street. Note also that 17221 and 22117, respectively, have the prime factorizations 17 * 1013 and 17 * 1301. [La Haye]

*Stalag 17* is a classic film set in a German prisoner of war (POW) camp. [Hartley]

A cow's saliva increases by 17% while grazing. [Patterson]

The reversal of 2^{17} is prime. [Patterson]

*17 Lectures on Fermat Numbers: From Number Theory to Geometry* was written in honor of the 400th anniversary of the birth of amateur mathematician and lawyer Pierre de Fermat. (17 is a Fermat prime.)

The smallest emirp for which both associated primes are the lesser in a twin prime pair. [Capelle]

According to hacker's lore, 17 is described at MIT as "the least random number."

The smallest automorphic prime, i.e., a prime p such that p is the kth prime and p ends in k. [Russo and Rivera]

The longest word of prime length in the King James Version of the Bible (the "KJV Bible") is *Chushanrishathaim* with 17 letters. This was the king of Mesopotamia and the first oppressor of the Israelites after their settlement in Canaan. [Opao]

!17 - 1 is prime. Note that !17 represents subfactorial 17. [Gupta]

17 is the largest number less than a thousand such that the sum of its factorial and primorial plus and minus one are primes. I.e., 17!+17#-1 and 17!+17#+1 are primes. [Opao]

The middle verse in the New Testament is Acts 17:17.

Vietnam was divided along the 17th parallel of latitude.

The smallest prime (emirp) equidistant both between two Mersenne primes (3 and 31) and between two perfect numbers (6 and 28). [Beedassy]

A prime formed from the leading 2 decimal digits of sqrt(3). [Carmody]

tan(cos(sin(x))) = .017... for any x, when this expression is computed in degree mode. [Dillon]

March 9, 2004, Senator Pat Roberts, Chairman of the Intelligence Committee, asks George Tenet, Director of the CIA, "Can you splinter a shingle 17 ways." (Paraphrased) [Haga]

The "*Encyclopedie*" was published in France with 17 volumes of articles issued from 1751 to 1765. Denis Diderot coedited the monumental work with Jean le Rond d'Alembert (born in 1717). [Marr]

The smallest prime that is the average of the squares of two primes (3 and 5). [Gallardo]

Billionaire Bruton Smith bought his first race car at age 17.

17 is the (1 * 7)th prime. [Firoozbakht]

The sum of the first 17 composite numbers is prime. [Patterson]

There were 17 episodes of the classic BBC TV series *The Prisoner*. [McCranie]

17 is the smallest prime of form x^2 + y^2 + z, where x, y and z are three successive positive integers (2^2 + 3^2 + 4 = 17). [Teofilatto]

17 is the only positive prime Genocchi number. [Terr]

The largest known prime that is not the sum of two semiprimes. [Manor]

17 was called by the Pythagoreans - opposition, obstruction, and evil, and the day the Devil triumphed over God.

St. Patrick, the Patron Saint of Ireland, is believed to have died on March 17. [Dowdy]

17 = 2+3+5+7 is the smallest emirp among sums of consecutive primes. Note that the reverse 71 = 19+23+29 is also such a prime. [Poo Sung]

There is an absence of Y-chromosome marker H17 in Polynesian populations. [Brookfield]

The first program to run on a stored-program computer
consisted of 17 instructions. It was used to find the largest
factor of 2^{30} - 1. [McCranie]

1/(sqrt(17) - 4) = sqrt(17) + 4 [Moody]

There are 17 words in the following quotation: "It is evident that the primes are randomly distributed but, unfortunately, we don't know what 'random' means." R. C. Vaughan (February 1990). [Post]

17 is the only prime number in existence that is equal to the square of a prime number plus 8. (17 = 3 * 3 + 8) No other prime number squared plus 8 results in a prime. [Goelz]

All whole distances up to 17 can be measured using a Golomb ruler bearing as few as six marks. [Beedassy]

In France, one dials 17 to call for Police Help in emergency. [Beedassy]

In poker games starting with two hole cards, one out of 17 starting hands is a pair. [McCranie]

There are 17 standard-form types of quadratic surfaces (quadrics). [Poo Sung]

Using the standard English set of integer names (i.e., not counting special names such as "googol"), 17 (seventeen) is the largest integer whose name is spelled with only one distinct vowel. [Brahinsky]

The sum of digits of the square of 17 is its twin prime. [Silva]

Ramanujan defined 17 Jacobi theta function-like functions which he called "mock theta functions" in his last letter to Hardy.

David C. Kelly, a math professor at Hampshire College in Amherst, Massachusetts, gives an annual lecture on the number 17.

!17 is the smallest subfactorial containing all digits from 0 to 9, i.e., pandigital. [Gupta]

"17" is the title of a song by Jethro Tull. [McCranie]

Only 17 original copies of the *Magna Carta* are known to survive. Texas billionaire and ex-presidential candidate Ross Perot once owned a copy.

The smallest multidigit prime which requires exactly a prime number of letters to spell in both English and Latin ("seventeen" has 19 letters, and "septemdecim" has 11 letters). [Post]

17 * 5882353 = 100000001, and the reciprocal of 5882353, rounded to 1+7 decimal places, is 0.00000017. Note that 5882353 is also a prime number.

The first prime that is not a Higgs prime for squares or cubes. This follows from the fact that 17 is a Fermat prime F_{n} = 2^{2n} + 1 (with n = 2), and hence not a Higgs prime for powers below 2^{n}. [Beedassy]

In Mauritian Creole rhetoric, the number 17 is used as a figure of speech to denote a pretty young lady. [Beedassy]

The Leonid Meteor Shower is visible every year around November 17 as pieces of debris associated with the comet Tempel-Tuttle enter Earth's atmosphere at some 71 kilometers per second.

There are 17 distinct sets of regular polygons that can be packed around a point. [Astle]

The National Football League's New York Giants won Super Bowl XLII by three points when player number 17 (Plaxico Burress) caught the game-winning pass to make their score 17. [Carnahan]

All calculus students should know basic trigonometric values such as cos(2π/3). But have you ever seen Gauss's formula for cos(2π/17)?

The smallest prime number which is not the difference of two Fibonacci numbers. Hence, it is the smallest prime number which is not the sum of consecutive Fibonacci numbers. [Capelle]

The smallest Leyland emirp. [Loungrides]

A standard Bb clarinet has 17 keys. [McCranie]

Apollo 17 was the last manned Moon landing space mission. [Newman]

"Edge of Seventeen" is a song by American singer Stevie Nicks, the third single from her successful 1981 solo debut album Bella Donna. [Montardit]

The largest known prime *p* such that 4*p*# + 1
is the square of a prime. [Beedassy]

10^17-3 and 10^17+3 are sexy primes. [Margus]

The smallest fortunate emirp number whose the reversal is also a fortunate number. [Loungrides]

The only known Mersenne prime exponent *p* such that *p*^{2} ± 2 and *p*^{3} ± 2 are semiprimes.

2*7*13 - 3*5*11 = 17 (first seven primes); Note that 17 (prime(7)) is of the form A - B where A*B = prime(1)*prime(2)*...*prime(6). Is 17 the largest such prime? [Firoozbakht]

The number of primes of Smarandache-Wellin type descending by increments from a power of ten that are smaller than the smallest one ascending by increments from a power of ten. [Merickel]

The smallest prime whose binary expansion (10001_{2}) contains a greater (prime) number of 0's than that of 1's. [Beedassy]

The Munich Oktoberfest is 17 days long when the 1st Sunday is October 2.

The smallest prime factor of the semiprime 17#-1 is the reversal of 17-1. [Post]

The largest known natural number that cannot be expressed as the sum of 2 or 3 distinct primes, but the smallest that can be expressed as the sum of 4 distinct primes. [Capelle]

A small prime price (List Price: $17.00) for a great, not-so-trivial, prime book: "Prime Curios! The Dictionary of Prime Number Trivia" by Chris K. Caldwell and G. L. Honaker, Jr. [Green]

The smallest strong emirp. Note that its reversal is also a strong emirp. [Loungrides]

The only Fermat emirp. [Loungrides]

1!! + 2!! + ... + 15!! = 16 (mod 17). It is the only known prime with this property. [Firoozbakht]

The only prime (emirp) such that 17-2d is a prime where d is any member of a sequence of four consecutive primes (case: 2, 3, 5, 7). [Loungrides]

The only emirp of the form p^2p+1 where p is a prime. [Loungrides]

In the James Bond movie "On Her Majesty's Secret Service," 17 is the number of Bond's private office.

"Prime Number: 17 stories from Illinois short fiction" serves an appetizer for those who wish to sample the hundreds of stories by individual writers.

The smallest emirp *p* such that the digits of all
primes up to *p* add up to an emirp: 2 + 3 + 5 + 7 + 1
+ 1 + 1 + 3 + 1 + 7 = 31. [Beedassy]

The smallest prime average of seven consecutive primes: 7, 11, 13, 17, 19, 23, 29. [Silva]

Emirp 17 yields three other emirps if 1, 7, and 1+7 zeros are sandwiched between its digits. Obviously, the same can be said of emirp 71. [Silva]

The number of different ways its emirp partner 71 converts into a prime from a base less than ten to a base greater than it up to and including ten (likely maximum regardless of whether the original is prime). [Merickel]

The smallest emirp that remains an emirp when each digit is replaced by its square: 149. [Beedassy]

The emirp number of letters in the classic metaphore
*"The atoms of numbers"* in reference to the primes.
Note that concatenating each word's number of letters forms
an emirp with each distinct prime digit: 3527. [Beedassy]

The smallest prime p such that p#+1 and p#-1 are both composite. [Poo Sung]

There are 17 capital letters in the English alphabet without horizontal symmetry. Can you find them all? [Delph]

The number 17 frequently signifies Heaven.

Subdermatoglyphic contains 17 letters and is the longest word in which every letter is used only once. It refers to the underlying dermal matrix that determines the pattern formed by the whorls, arches and ridges of our fingerprints. [Homewood]

It has been proven (January 2012) by Gary McGuire and his collaborators of University College Dublin that 17 is the fewest number of clues that can be provided in a Sudoku puzzle that will lead to a unique solution. [Green]

Chess Grandmaster and friend of Bobby Fischer, Robert Byrne, accurately predicted that World Champion Boris Spassky would get 17 half points in his match with Fischer. [Merickel]

The only double-digit prime whose cube has sum of digits equal to p. [Loungrides]

The number of letters in 'RIEMANN HYPOTHESIS'. [Homewood]

17 = 2^3 + 3^2. [Bui Quang Tuan]

The adult brown marmorated stink bug (Halyomorpha halys) is approximately 17 millimeters long. [Fields]

The only non-titanic prime of form 2^p+p^2 where p is a prime, (p=3). [Loungrides]

Math Salamaders know primes up to 17.

17 is the least possible number of moves to win a BINGO game. [Moorthi]

The 17th letter of the alphabet, Q, is the only letter that doesn't appear in the name of any U.S. state. [Gaydos]

There are 17 happy two-digit numbers. [Gaydos]

17 is the only prime number that is half of a Fibonacci number (34). [Jacobs]

On a TRS-80 Model 1 floppy disk, the directory was on track 17. [Gaydos]

In Islamic prayer there are 17 compulsory Rakats (groups of units of prayer) per day. [Jiffrey]

Six times the natural logarithm of 17 is almost 17. [Gaydos]

Prime numbers that end with "17" occur more often than any other 2-digit ending among the first 100 million primes. [Gaydos]

There are exactly 17 odd primes with distinct prime digits. [Khovanova]

From the sum of the first n primes one notices that from n = 2*17 to 2*21, all five end with the digit 7; the same as 17. The sums are 2127, 2427, 2747, 3087, 3447. [Bergot]

The Lo Shu (雒書) is a unique magic square that contains exactly 17 primes according to word search rules. Can you find them all? [Honaker]

Q is the 17th letter of the modern English alphabet. If A = 1, B = 2, C = 3, ... , Z = 26, then Q=U+E+E-N. Found by Susan Thorpe of Great Missenden, Buckinghamshire, England.

"What Great Teachers Do Differently: 17 Things That Matter Most" was written by Todd Whitaker, who taught mathematics early in his career.

The sum of the digits of first primes up to emirp 17 is the next emirp. [Silva]

If in each of the twin primes 17 and 19 you sandwich the last digit 9, you will obtain a pair of three-digit twin primes (197, 199). [Silva]

The fiftieth known Mersenne prime exponent (discovered in '17) ends in 17. Note that the Mersenne prime itself ends in the reversal of 17. [Wang]

Subtract all the primes less than 17 from 17 and add the results if they are semiprimes. The result is the square of a prime. [Bergot]

3+14 is a prime number.

17^{7}+17^{4}+17^{1}+17^{3}+17^{2}+17^{4}+17^{5}+17^{6}+17^{5}+17^{8} = 7413245658. [Vosbard]

17 is the minimal prime solution as sum of three distinct triangular integers: 17=1+6+10. [Rivera]

The sum of all prime digits. [Gupta]

There are at least 17 species of ragweed that grow in North America. [Ellis]

The eastern United States is bracing for an onslaught of millions of noisy 17-year cicadas in 2020. [Dolasia]

The smallest prime that is not the sum of two Fibonacci numbers. [Honaker]

The smallest prime congruent to 3 (mod 14). [Honaker]

The book "Ramblings of an Old Man" by Michael Heffron describes the significance of prime number 17 through history.

Floyd Collins' dead body was finally reached 17 meters below ground level on February 17. [Strickland]

Traditional Japanese haiku consist of phrases composed of 17 phonetic units. [Toyama]

17! is the smallest factorial number that falls between two sphenic numbers. [Gaydos]

There are exactly 17 different prime numbers in the substrings of the digits of 17^17. [Gaydos]

Position 17 in the decimal expansion of both π and *e* contains the same prime digit. [Gupta]

The word *primogenitureship* contains 17 letters and first
appeared in English in the 17th century.

Calculate the sum of the emirp 17 and its reversal. The sum of all prime factors of this number is 17. [Bergot]

The sum of the first 6 (and perhaps only known) digits of the Leviathan number. [Olry]

2*3*5*7+11=13*17. Note that all the primes up to 17 are used. [Hollesen]

Is n = 17 the only n^n that contains exactly n distinct prime numbers in the substrings of its digits? [Gaydos]