# 7

This number is a prime.

There are 7 letters in TUESDAY, which is the only day of the week whose name contains a prime number of letters. [Gupta]

The first prime number of the form 6*n* + 1. All primes greater than three have either the form 6*n* + 1 or 6*n* - 1.

Graph theorists have been able to prove that 7 colors are required on a donut-shaped map (i.e., an ordinary one-holed torus) to ensure that no adjacent areas are the same.

The first 7 digits of 8^{9} form a prime. [Kulsha]

The only single digit prime number to yield a square when its cube and the cube's proper divisors are added. Thus 7^{3} + (1 + 7 + 7^{2}) = 20^{2}.

7! - 6! + 5! - 4! + 3! - 2! + 1! is prime. [Guy]

Seven is the only odd prime that becomes "even" by deleting a letter. [Beedassy]

There are only 7 smoothly undulating nine-digit palindromic primes.

"Seventh heaven" is the farthest of the concentric spheres containing the stars in the Muslim and Cabalist systems.

Pill bugs ("roly-polies") have 7 pairs of legs.

In 1992, Bayer and Diaconis showed that after 7 random riffle shuffles of a deck of 52 cards, every configuration is nearly equally likely.

The Seven Bridges of Königsberg is a famous solved mathematics problem inspired by an actual place (now Kaliningrad, Russia) and situation. [Millington]

The smallest odd full-period prime.

If you turn a U.S. one-dollar bill upside down and place your thumb over the eagle's head, the shield becomes the Menorah, or the 7 candlesticks of Israel.

Jesus uses the word seven a "perfect number" of times in the last verse of the first chapter in the Book of Revelation. This makes perfect sense because seven is the number of perfection in the Bible. [Zirkle]

The only odd prime that solves 6*y*^{2} = (*x* + 1)(*x*^{2} - *x* + 6). [Ljunggren]

The first prime happy number. Replace a number by the sum of the squares of its digits, and repeat the process until the number equals 1, or it loops endlessly in a cycle which does not include 1. Those numbers for which this process ends in 1 are called happy numbers. There are exactly 7 prime happy numbers less than 100. Can you find all 7?

There are 7 indeterminate forms involving 0, 1, and infinity.

7 Up (or Seven Up) is a brand of a lemon-lime flavored soft drink.

1^{7} + 4^{7} + 4^{7} + 5^{7} + 9^{7} + 9^{7} + 2^{7} + 9^{7} = 14459929. [Randle]

Thabit ibn Qurra (circa 850 A.D.) noted that if n>1 and each of p = 3*2^(n-1) - 1, q = 3*2^n - 1, and r =(p+1)*(q+1) - 1 is prime, then (2^n*p*q) and (2^n*r) are amicable pairs. Descartes wrote to Mersenne in 1638 with the pair 9363584 and 9437056 (n = 7).

Add 7 to 720(100^{49} - 1)/99 to get a 99-digit undulating prime. [Pickover]

The largest known number of consecutive primes that can be partitioned into two sets such that the difference of their product is unity: (5 * 11 * 13) - (2 * 3 * 7 * 17) = 1.

The Clay Mathematics Institute announced in May 2000 a prize of $1,000,000 for each of the 7 unsolved Millennium Prize Problems. The Poincaré conjecture (essentially the first deep conjecture ever made in topology) has been the only Millennium problem solved thus far.

If asked to quickly pick a number between 12 and 5, most people will choose 7.

If *n* is composite, the first 7 primes of the form sigma(*n*)/*d*(*n*) are Mersenne prime exponents.

Michael Bell proved that 10^(100**n*) + *n* is prime, where *n* = 7. As of May 2000,
it is the only known prime of this form.

7! + 19^{19} is a prime number with 19's on both ends. [Kulsha]

There are 7 different notes in a standard major scale in music as well as in a standard minor scale. [Obeidin]

All primes p greater than or equal to 7 divide a concatenation of (p-1) ones. [Murthy]

Using A = 1, B = 2, ..., Z = 26, the sum of all the letter values in the word names of the numbers from 1 to 7 is a prime (367). If 0 is included, the sum is yet another prime (431). [Trotter]

Double 7! to get the exact number of minutes in a week (7 days). [Luhn]

The 'Tractatus Logico-Philosophicus' of Ludwig Wittgenstein brings 7 groups of aphorisms. The 7th group brings only one (a very curious one): Wovon man nicht sprechen kann, darüber muss man schweigen. About what one cannot speak, one must keep silent. [Gevisier]

A Mersenne prime (2^{p} - 1) with exponent *p* = 3. The key prime factor of the perfect number 28. There are twenty-eight 3-digit numbers with digits summing to 7 (e.g., 601; 412) among which exactly 7 (e.g., 241; 313) are prime and also among which one (511) is of the form 2^{q} - 1 where *q* = 3 * 3. The seven 3-digit primes sum to 2281, a prime with an embedded "28" and with first and last digits summing to 3. Note that the number 2281 is also the exponent of the 17th Mersenne prime.

The reversal of (7 - 1)! + 1 is 2^{7} - 1 (a Mersenne prime). [Luhn]

There are "7 deadly sins" used in early Christian teachings to educate and protect followers from basic human instincts (pride, envy, gluttony, lust, anger, avarice, and sloth). [Croll]

The first number of every sextuplet ends with 7. [Luhn]

The sum of the digits of each of the first two multidigit squares is 7. [Sladcik]

The smallest prime of all integers n such that Mordell's equation y^2 = x^3 + n has no integral solutions. [Post]

There were 7 rings of power for the Dwarf-lords in the stories of J.R.R. Tolkien. [Oldenbeuving]

The number of points and lines on the minimal finite projective (or Fano) plane. [Poo Sung]

It is possible to place 7 cigarettes in such a way that each touches the others if the length divided by the diameter of each is greater than or equal to 7*sqrt(3)/2. [Gardner]

De Morgan suspected Shanks's calculation of π up to 707 digits was wrong because of the lack of 7s contrary to other digits. [Poo Sung]

(77 + 777)/(7 + 7) = 61 is prime. Note that seven 7's are used and 6 + 1 = 7. [Poo Sung]

There were 7 papers in the first edition of the Ramanujan Journal. [Croll]

There are 7 seals in the New Testament's Book of Revelation. [Croll]

There is a persistent rumor that the philosopher G.W.F. Hegel (1770-1831) provided a logical proof within his doctoral dissertation that there could only be 7 planets in the solar system. [Poo Sung]

The only prime whose sum of divisors (p+1) is a cube. [Gupta]

The smallest prime which can be expressed as the sum of a prime and a composite number (7 = 3 + 4). [Murthy]

In most Hindu marriages the bride follows the groom 7 times around the holy fire, which is called a Saptapadi. [Das]

The smallest example of x-y,xx-y,xxx-y all being prime is with digits x=7 and y=4. So 7-4=3, 77-4=73, 777-4=773 are all primes. It is the only solution because one of three consecutive numbers x..x-y is divisible by 3. [Luhn]

(2^7-1)/(2-1) and (2^7+1)/(2+1) are both primes. [Dobb]

Uranus' moon Miranda (as viewed from space) has a "7" embedded in the middle of a rectangular corona that appears to have been formed by viscous icy lavas. Uranus is the 7th planet from the Sun.

There are 7 primes less than 77 that have at least one digit being 7 (7, 17, 37, 47, 67, 71, 73). [Bush]

7 is the smallest integer that is not the difference of two primes. [North]

The very first Menorah for the Tabernacle as commanded by God to be built in the book of Exodus had 7 lamps in it. [Obeidin]

The only prime such that all the positive values of 7-2^k are all primes, where k is not equal to 0. [Russo]

Seven Segment Display |

The smallest Woodall prime. [Russo]

The smallest cuban prime. The name has nothing to do with Cuba the country.

The bluntnose sevengill shark has 7 long gill slits in front of each pectoral fin. [Jolly]

7 = sod(sod(sod(4444^4444))), where sod(n) is the sum of digits of integer n. [Daly]

"Seven Minutes of Terror" is the phrase given to Curiosity's final minutes to landing on Mars. Touchdown on the floor of Gale Crater is expected at 1:17 a.m. (EDT) on 6 August 2012.

The Japanese word "Subaru" refers to an open star cluster called the Seven Sisters (Pleiades).

Kwanzaa (or Kwaanza) is a 7-day festival (December 26 to January 1) celebrated primarily in the United States, honoring African-American heritage. [Karenga]

The number of stars that form the asterism known as the Big Dipper in North America.

Roma was built over 7 hills. [Franco]

The only prime that can be the digital root (sum of digits computed recursively until one digit remains) of a perfect square. [Rupinski]

13 + 17 + 19 = 7^2. [La Haye]

Andrew Wiles spent 7 years working on his solution of Fermat's Last Theorem. [Croll]

7 is the only prime defined as a cube minus one. [Gallardo]

In "Western" music's 12-tone scale, ratios of fractions between 2 and 1 with a one-digit denominator that involve 7 are not used. This is true whether 7 is either in the numerator or denominator. Ratios such as 7/4, 7/5, 7/6, 8/7, 9/7, etc. are often common notes in the music of other cultures or blues music, but not the "Western" equal temperament scale. [Obeidin]

The maximum number of eclipses of the Sun and Moon that can occur in any one year. [Byrd]

7 is the number of frieze groups, i.e., the groups consisting of symmetries of the plane whose group of translations is isomorphic to the group of integers.

In 1995, Ramaré showed that all integers greater than one are the sum of at most 7 prime numbers.

The number of distinct prime knots containing 7 crossings. [Tait]

7^{2} can be written as a sum of squares in 7^{2} + 7^{2} ways. [Hartley]

The smallest sexy prime triplet that is not part of a sexy prime quadruplet or quintuplet starts at 7. I.e, 7+6 and 7+6+6 are also prime. [Opao]

7^12 reversed is prime. [Patterson]

The numbers on opposite sides of a standard die always add up to 7.

The number of rank and good ears of corn that came up upon one stalk in Pharaoh's second dream (Genesis 41:5).

The sum of seven consecutive primes beginning with 7 is seven times the seventh prime. [Post]

The number of visible colors formed by light across a prism. [Gallardo]

According to the History Channel - 7 machine tools started the modern age: Lathe, Planer, Mill, Grinder, Drill, Band Saw, and Punch-Press. [McAlee]

The first exotic differentiable structure was a 7-dimensional hypersphere. It was found by Milnor (1956) using the calculus of octonions. [Poo Sung]

1!*2!*3!*4!*5!*6!*7! ± 1 are twin primes. [Gupta]

Atop the *Statue of Liberty* there are seven spikes, representing the seven seas. [Patterson]

sod(M4 exponent) = sod(M9 exponent) = sod(M16 exponent) = 7. [Harrison]

7 of the major planets in our solar system have natural satellites. [Wolfe]

7 is the smallest prime number that is equal to the product minus the sum of twin primes, i.e., 3*5 - (3+5) = 7. [Teofilatto]

The smallest prime p such that all five numbers p*10^k+1, k < 6 are primes. [Firoozbakht]

7 is the smallest Carol prime. [McAlee]

7 = 2^5 - 5^2. [Astle]

7 is the only number n such that n = π(n)! - prime(n). [Firoozbakht]

Levy's conjecture (1963) states that all odd numbers greater than or equal to 7 are the sum of a prime plus twice a prime. It was named after Hyman Levy, who was apparently unaware that the conjecture was first stated by Émile Lemoine in 1894. [Capelle]

7 is a double Mersenne prime. [Post]

7 is the first centered hexagonal prime. [Post]

There are '7 things you need to know about prime numbers.' - Dr. Vicky Neale

The smallest prime followed by a prime number of consecutive composite numbers. [Silva]

7 is the smallest number that cannot be represented as a sum of 3 squares. [Poo Sung]

Assuming Goldbach's conjecture, there is no positive number that can be written as the sum of exactly 7 primes in exactly 7 ways. [Hartley]

Have you ever noticed that the word "indivisibilities" contains 7 i's?

The smallest prime whose reciprocal has semiprime period length. [Post]

Whilst every Fibonacci prime number has odd prime subscript, 7 is the first prime not appearing in the Fibonacci sequence. [Beedassy]

John Elway, one of the most formidable figures in NFL history, wore number 7 for the Denver Broncos.

The smallest prime NSW number. [Beedassy]

There are exactly 7 possible basic shapes for six-sided polyhedra (convex hexahedra). [Caldwell]

7 is the only prime in which a/b + c/d + e/f = a/b * c/d * e/f, where the variables a through f represent any nonzero single-digit numbers. [Morse]

The smallest hyperfibonacci prime that is not a Fibonacci prime. [Post]

Ladybugs have 7 spots on their back. [McCranie]

Divisibility test for 7 (or 13): Combine the digits in order into groups of 3 (starting from the right) by alternating them with positive and negative signs. If the result is divisible by 7 (or 13 respectively), then so is the original number. For example, 7 divides 62540982 because 7 divides +(62)-(540)+(982).

The smallest prime p that yields a number of the form pp when multiplied by the next prime. [Capelle]

Insert 7 in the middle of the 7th power of 7 and you will get a 7-digit prime. [Silva]

7*6*5*4*3*2*1 + 7654321 is a 7-digit prime with first digit 7 and 1*2*3*4*5*6*7 + 1234567 is a 7-digit prime with last digit 7. [Silva]

The only prime p such that
π(p^{2}) <
(π(p))^{2}. [Capelle]

Patterson's Conjecture: If P_{n} is the nth prime, then the
series (P_{1}-1 + P_{2}-2 + P_{3}-3 + ... + P_{n}-n) is divisible by Pn if and only if P_{n} = 7.

It is possible to construct an angle of π/7 using 7 matchsticks. [Beedassy]

Although baseball's first World Series, in 1903, was an 8 game series, as was 1912, 1919, and 1921, it has ever since been a best-of-seven-game series. [Post]

The dimensions of irreducible modules of the exceptional Lie group G_2 are all congruent to {-1,0,+1} mod 7. A similar phenomenon occurs for all exceptional Lie groups and for some classical ones as well. [Rupinski]

The smallest prime subscript *n* such that
*p*_{n+1} is the first prime dividing
#*p _{n}*+1, (

*p*being the

_{n}*n*th prime ). [Beedassy]

The only prime equal to the difference of the product and the sum of the two previous primes. [Silva]

The normal human head bears 7 sense organ openings: 2 eye sockets, 2 ear canals, 2 nostrils, and a mouth. [Beedassy]

The only prime that is member of two pairs of cousin primes. [Silva]

The 7 heavenly virtues are prudence, temperance, justice, fortitude, charity, hope and faith. [Croll]

The Hamming-7 code is a perfect error-correcting code of word-length 7, with 4 message carrying bits and 3 error-correcting bits. [Sirag]

The game *Tetris* uses 7 one-sided tetrominoes (referred to as I, J, L, O, S, T, and Z). [Poo Sung]

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

7 = 98532/14076 is the unique pandigital fraction expression including 0. [Poo Sung]

The only odd prime *p* whose *n*-th power
*p*^{n}, for any nonzero *n* below
*p*, has exactly *n* digits. [Beedassy]

The checksum of the checksum of the first 7 decimal places of the common logarithm of 7 is 7 (log 7 = 0.8450980, 8+4+5+0+9+8+0=34, = 3+4). [Vastvision]

There are 7^{2} natural numbers n
such that n is equal to the number of 7's in the decimal
digits of all numbers smaller or equal to n. [Capelle]

The smallest prime whose square is a concatenation of consecutive prime squares. [Silva]

The largest prime number whose third power (343) can be displayed on a 12-hour digital clock. [Fellows]

'Oh Thank Heaven for 7-Eleven'. The 7-Eleven (R) name reflects the stores' extended hours - 7 a.m. until 11 p.m., seven days a week. July 11 (7/11) is the official birthday of the 7-Eleven convenience store chain. [Green]

The smallest prime with a composite number of partitions. [Pol]

The Pollock octahedral numbers conjecture (1850) states that every number can be written as the sum of at most seven octahedral numbers. Frederick Pollock, better known as a lawyer and politician, was a contributor of papers on mathematics to the Royal Society. [Capelle]

The number of distinct four of fewer face-joined unit cubes forming the polycube pieces of the 3-by-3-by-3 Soma cube dissection puzzle invented by Piet Hein in 1936. [Beedassy]

The largest known prime number that is not the sum of a triangular number, a square and a cube, all of them greater than or equal to 1. [Capelle]

Seven prime numbers cannot be written as a sum of squares of primes. Note that 7 is one of them and divides their sum. [Capelle]

The largest known prime number p for which the decimal
expansion of p^{p} does not
contain any pair of consecutive equal digits. [Capelle]

There are 7 known natural numbers n such that n is the sum of two consecutive primes but not the sum of two sums of two consecutive primes. [Capelle]

There are 7 improper zeroless pandigital fractions that equal 7. For example, 37926/5418 = 7. [Bolling]

All multidigit Fermat numbers end in 7. [Gupta]

Dubai eGovernment has a logo in line with law number 7 of 2009 issued by H.H. Sheikh Mohammed Bin Rashid Al Maktoum, Vice President and Prime Minister of the UAE and Ruler of Dubai, which is one of the 7 emirates and the most populous state of the United Arab Emirates.

The number of capital letters of English alphabet with no symmetry (F, G, J, L, P, Q, R). Note that the letters P, Q and R are often used to represent the primes. [Capelle]

The smallest prime p followed by more than one prime prior to 2p. [Silva]

The number of strobogrammatic capital letters of the English alphabet (H, I, N, O, S, X, Z). The word strobogrammatic begins with S, the only strobogrammatic letter with a prime index. [Capelle]

"The Seven Year Itch" (1955) is a highly-rated comedy film starring Tom Ewell and Marilyn Monroe. The titular phrase refers to a declining interest in a monogamous relationship after seven years of marriage. [Green]

All 7 numbers n! + nextprime(n), n < = 7 are primes. [Firoozbakht]

Each hind foot of Ichthyostega (one of the first tetrapods) had 7 toes. [Lindsay]

A segment called "Seven Prime Questions" first aired on PBS in January 2006.

The sum of all untouchable primes. [Loungrides]

"The time will come when diligent research over long
periods will bring to light things which are now hidden." -
Seneca, *Natural Questions*, Book 7, first century

1! + 2! + 3! + 4! + 5! = 6 (mod 7). It is the largest known prime (up to 10^5) with this property. [Firoozbakht]

The only prime, p, such that σ(p^{3}) is a square. [Wesolowski]

Every connected (non-empty) one-dimensional topological manifold is homeomorphic to one of these 7: the circle, the closed interval, the open interval (real line), the half-open interval, the closed long ray, the open long ray, or the long line. [Post]

The smallest of only two primes less than a titanic prime of the form p^p+p+1, where p is a prime. The other is 31. [Loungrides]

7 is the 3rd lucky number. It is also the rank of the E7 Lie algebra (E6 and E8 are the only other E-type Lie algebras). [Sirag]

There are 7 hidden dimensions in 11-d Supergravity, which is the "low energy" approximation to M theory, which also has 7 hidden dimensions. [Sirag]

The metric system uses 7 fundamental units of measure: m, kg, K, mol, A, sec, and cd. [Jauregui]

The Heawood graph corresponds to the 7-color torus map on fourteen nodes. Note that "Heawood" has 7 letters. [Post]

The only two-syllable prime digit in English. [Mia]

The only known number whose factorial has 50% of its digits consisting of 0's: 7! = 5040. [Beedassy]

The smallest number n that yields two primes when preceded or followed by n ones. [Silva]

H-7 is the heaviest isotope of hydrogen observed experimentally in a heavy particle accelerator. [Beedassy]

"Seven Years in Tibet" is an American biographical war drama film based on the book of the same name. [Galliani]

The first 7 odd primes are all members of twin prime pairs. [Silva]

The French version of the popular cartoon *Spot the difference* puzzle is called "*Le jeu des 7
erreurs*" (meaning 'The 7 mistakes game'). [Beedassy]

Largest known prime which is the sum of first n Fibonacci numbers (case n=4), i.e. 7 = 1+1+2+3. [Gupta]

7! + 6! - 5! + 4! - 3! + 2! - 1! and 7! - 6! + 5! - 4! + 3! - 2! + 1! are each members of a twin prime pair! [Homewood]

(7, 11, 13) is the first consecutive prime number triple (p, q, r) such that r divides pq+1. Note that the quotient is 6 (the first perfect number). [Loungrides]

There are 7 sexy prime quadruplets up to a thousand. [Homewood]

Carl B. Pomerance received his Ph.D. from Harvard University with a dissertation proving that any odd perfect number has at least 7 distinct prime factors.

Giraffes have seven neck bones, which is the same as humans and most other mammals but longer. [Capelle]

Why is 10 afraid of 7? Because 7 8 9 and 10 is next! [Homewood]

*Avenged Sevenfold* (sometimes abbreviated as A7X) is an American heavy metal band from Huntington Beach, California. [Allison]

The only prime heptagonal number. [Homewood]

The book of Genesis describes the process of creation from cosmic chaos to cosmic order in 7 stages (days). [Green]

The smallest possible number of spins necessary to win a game of Chutes and Ladders. [Goelz]

(123465789 + 987654321) +/-7 are both prime. [Bopardikar]

7 squared is the sum of 7 consecutive integers. [Silva]

The number of ways that primes can be written as the sum of one or more distinct natural numbers less than or equal to 4. That is to say {2}, {3}, {1+2}, {1+4}, {2+3}, {3+4}, {1+2+4}. [Deutsch]

A mnemonic for remembering the first seven primes is, "In the early morning, astronomers spiritualized nonmathematicians."

π(Look-and-Say (7)) = 7. [Honaker]

*An Encyclopedia of Seven* is a fascinating and fun-filled compendium celebrating all things relating to the number 7.

The only non-titanic prime of form n^n+3, i.e., 2^2+3. [Loungrides]

Kaprekar's routine will always reach its fixed point in at most 7 iterations.

The average score for this Prime Numbers quiz is greater than or equal to 7.

The pH value of pure water is 7. [Gupta]

"Shmita" also called the sabbatical year or shǝvi'it (שביעית, literally "seventh") is the seventh year of the seven-year agricultural cycle mandated by the Torah for the Land of Israel.

If p is a prime greater or equal to 7 then any repunit number consisting of p-1 one’s is divisible by p. [Loungrides]

There is only one known case where the average of an amicable pair sum ends with the digit 7. The pair (1184, 1210) was discovered in 1866 by teenager B. Nicolò I. Paganini.

Bell's palsy is a form of temporary facial paralysis resulting from damage or trauma to facial nerve number 7.

7 is a rare digit at the beginning of π. [Jacobs]

The remainder is 7 when sum of the first 7 primes is divided by the 7th prime. Is this the only odd prime example? [Bergot]

The population standard deviation of the first 7 natural numbers is "odd."

In 1956, psychologist George A. Miller published a paper in
*The Psychological Review* in which he talks about the
prime number 7 "following him around."

The sum of all n^k with n + k = 7, i.e., 1^6 + 2^5 + 3^4 + 4^3 + 5^2 + 6^1 + 7^0, = 7#. [Bergot]

The only twin prime of the form 30n-23. [Homewood]

The first half of a leap year and the last half of any year can, only in different locations, have a number of hours divisible by 7. [Merickel]

Is 7 the only prime area for an n-gon plotted (n, prime(n))? Start (1,2), (2,3), ... ,(8,19) and connect (1,2) to (8,19). [Bergot]

Seven (stylized as SE7EN) is an American neo-noir crime thriller film directed by David Finche.

The standard ASCII (*American Standard Code for
Information Interchange*) character codes for encoding
symbols on a typical American keyboard were originally 7
bits long. [Beedassy]

7+10^999 is the smallest titanic prime that remains prime after you drop all the even digits. [Rivera]

"The Diabolical Long Division Brain Teaser" involves the digit 7 and a bunch of asterisks.

The last digit of Graham's number, and the rarest digit among the 500 rightmost decimal digits of Graham's number (these 500 digits include 56 0s, 56 2s, 56 9s, 55 5s, 54 8s, 49 1s, 47 3s, 46 4s, 46 6s, but only 35 7s). [Olry]

The largest undeveloped uranium deposit in the United States (#7 in the world) is located in Pittsylvania County, Virginia. [Shulz]

The sum of prime numbers up to 7 is the 7th prime. [Wiszowaty]

7! + 7654321 is a 7-digit prime. [Silva]

Mark Spitz set an Olympic record in 1972 by winning 7 gold medals. [Homewood]

The Theory of Relativity can be explained in 7 minutes.

The smallest prime such that the previous two primes AND the next two primes are twin primes. [Gupta]

Number Theorist James Maynard received an award by The European Mathematical Society for his 2016 paper in which he proved there are infinitely many primes devoid of any one of the digits, and he used the rare property of primes without digit 7 as a test case. [Beedassy]

The only known centered octahedral prime. [Wiszowaty]

It is impossible to arrange 7 circles of the same diameter so that they each 'kiss', or touch, a central circle, also of the same diameter. [Collins]

(3+4)³ = 343 is a curious palindromic cube of 7. [Leonardis]

In a list of numbers of the form 3n + 1, starting with n = 0, the Mersenne primes have a Wagstaff prime index. [Homewood]