41 (number)

General Properties

• 41 - natural, odd , two-digit (decimal) number
• the square of 41 is 1681
• 2 ⁴¹ = 2 199 023 255 552
• 1/41 = 0.0243902 ... (infinite decimal fraction with a period of length 5)

Ways to represent a number

This section contains properties related to the questions of representing the number 41 as the values ​​of some remarkable numerical expressions.

• The 5th centered square number , that is, representable as the sum of two consecutive squares of natural numbers (41 = 4 ² + 5 ² );

also representable as the sum of three squares of natural numbers (41 = 6 ² + 2 ² + 1 ² ); while 41 is the smallest number, representable as the sum of squares of two natural numbers and at the same time as the sum of squares of three natural numbers, where all five of these numbers are different

• 41 = 21 ² - 20 ² , not representable in other ways as a difference of degrees (from 2) of two natural numbers
• 8th Prota number , 5th Prota prime number (41 = 5 × 2 ³ + 1)
• smallest positive integer not representable in the form | 2 ^x −3 ^y |, where x and y are natural numbers

Properties related to simplicity and divisibility

This section lists the properties of the number 41, related to the ratio of divisibility on the set of natural numbers.

• The 13th prime , the twin of the prime 43 , differs by six from the prime 47
• 6th Pythagorean prime number (looks like 4n + 1); 2nd (after 17) a prime number of the form 8n + 1
• 7th positive integer n such that n! +1 is prime ^[1]
• 5th strong prime
• 11th regular prime number
• 2nd Newman-Shanks-Williams Prime
• 7th number Sophie Germain (41 × 2 + 1 = 83 , which is also a prime number) ^[2] . The full sequence of Kunningam of the 1st kind of length 3 begins with the number 41 (41, 83, 167)
• the sum of the first six prime numbers: 41 = 2 + 3 + 5 + 7 + 11 + 13; also representable as a sum of three consecutive primes: 41 = 11 + 13 + 17
• equal to the sum of all positive divisors of numbers from 1 to 7: 41 = 1 + 1 + 2 + 1 + 3 + 1 + 2 + 4 + 1 + 5 + 1 + 2 + 3 + 6 + 1 + 7
• is present in Euler's trinomial (n ² + n + 41), the value of which is a prime number for integer n modulo less than 40
• the smallest prime number is such that the sum of all primes that do not exceed it is decomposed into the product of three different prime factors (238 = 2 × 7 × 17)
• the values ​​of certain number-theoretic functions:

Euler functions : φ (41) = 40;

pi functions : π (41) = 13;

τ-functions (the number of natural divisors): τ (41) = 2;

σ-functions (the sum of the natural dividers): σ (41) = 42 (41 is not enough );

Möbius functions : μ (41) = −1;

Mertens functions : M (41) = −1

Decimal notation properties and signs of divisibility

Here are the properties relating to the recording of the number 41 in the decimal number system and related issues (in particular, the convenience of oral computation and checking the divisibility).

• the decimal notation of the number 41 is formed by successively written two squares (4 and 1); its square 1681 has the same property (16 and 81)
• the number 99999 is divisible by 41 (10 ⁵ = 100000 gives the remainder 1 when divided by 41). Using this property helps to solve some problems of the Olympiad type . This property also implies the sign of divisibility by 41: “A natural number is divided by 41 if and only if, when separating all groups of five digits from its decimal record, starting from the end, and summing all the resulting natural numbers, a multiple of 41 is obtained” . (Example: the number 864694512870 is divisible by 41, since 86 + 46945 + 12870 is divisible by 41.)

The divisibility of the number 10 ⁵ −1 by 41 has several important consequences:

1. 41 divides the number 11111, as well as any number of the form 10 ... 010 ... 010 ... 010 ... 01, where in all four groups the same number of zeros does not equal 5k-1 with natural k; a stronger statement is also true: if any number, the number of characters in which is not a multiple of 5, is written on the line five times without spaces, then a number multiple of 41 will be obtained.
2. If a five-digit number is divisible by 41, then for any circular permutation of its digits, this divisibility is preserved (example: 16359 divided by 41, therefore, the numbers 63591, 35916, 59163 and 91635 are divided by 41). This fact is repeatedly indicated as an interesting property of the number 41 both by mathematicians ^{[3] [4]} and by amateurs of numerology ^{[5] [6]} .

• Other signs of divisibility by 41:
  • a number is divisible by 41 if and only if the modulus of the difference in the number of tens and fourfold number of units is divisible by 41. For example, 369 is divisible by 41, since${\displaystyle <span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">|</span></span>\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">36</span></span>}<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">-</span></span>\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">four</span></span>}<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">\cdot </span></span>\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">9</span></span>}<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">|</span></span><span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">=</span></span>\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">0</span></span>}}$ {\ displaystyle \ left | 36-4 \ cdot 9 \ right | = 0}   divided by 41.
  • To check whether a number is divisible by 41, it should be split from right to left into faces of 5 digits each. Then, in each face, multiply the first digit on the right by 1, multiply the second digit by 10, the third by 18, the fourth by 16, the fifth by 37 and add all the resulting works. The result will be divisible by 41 if and only if the number itself is divisible by 41 ^[7] .
• 41 is the smallest number such that the minimum value of the sum of a multiple of it is 5 ^[7]
• 41 is equal to the sum of the digits of the fifth degrees of the numbers 29, 56, 83, 92; 41 is not equal to the sum of the digits of a natural number to the power of 2 or 3 or 4

Other properties

• 41 ² = 40 ² + 9 ² (41 is the 7th number, which is the largest in some primitive Pythagorean triple ^[8] )
• 41 - 7th real Eisenstein prime number
• among the numbers from 1 to 40, quadratic residues modulo 41 are the numbers 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39 , 40, by quadratic nonresidues - the remaining numbers

Chemistry

• Atomic number of niobium

Physics and Astronomy

• Daphne is a large asteroid of the main belt , discovered by a telescope in 1856 by a Parisian astronomer .
• Stars 41 Lynx , 41 Cancer , 41 Snakes , 41 Altar , HAT-P-41 , WASP-41 ; cluster M41 .

Technical objects:

• SLC-41 (LC-41) - the launch pad at Cape Canaveral (USA), which has been launching spacecraft for major scientific missions for a long time, including Viking-1 and Viking-2 (Mars explorations), Mars Science Laboratory (delivery of the Curiosity rover to Mars), Voyager-1 (study of Jupiter and Saturn, determination of the boundaries of the Solar System), Voyager-2 (study of Uranus and Neptune, determination of the boundaries of the Solar System), New horizons (studies of Pluto, Charon).
• Site No. 41 of the Baikonur Cosmodrome (Kazakhstan, formerly the USSR) is known in connection with one of the largest catastrophes in the history of space flight .
• In 2014, 41 spacecraft were launched to explore the planet Mars .

Biology

• 41 ° C is the average body temperature in a number of animals (primarily birds) ^[9] . In humans, such a temperature is possible with a serious illness and is considered a dangerous sign ^[10] .

The significance of the number 41 in religion and mythology (legends, myths, rites) is largely related to the fundamental value of the preceding natural number 40, which is repeatedly found in all the most common religions (for example, several times in the biblical legend of the Flood , describing the life of the Prophet Mohammed etc.).

Inca

According to some chroniclers, the Inca calendar used a cycle consisting of 328 days, consisting of 41 weeks, 8 days each ^[11] .

Marians

Among the Mari, the number 41 plays an important role in many rites:

• At the wedding, the newlyweds must step on 41 evils and accept 41 kindness;
• At a funeral in a coffin to the departed, they put a knapsack in which there are 41 nuts ^[12] , and the person who replaces the departed departs from a certain place by 41 steps;
• During the holiday Semyk during the ritual campaign in the bath relies to brighten up your body with a broom connected from 41 species of plants.

Russian

In medieval Russia, the finger count was widely distributed among the “forties”: from the finger joints from one to forty. This method of counting means the actual use of the 40-ary number system , in which 40 is the largest digit and 41 is the first two-digit number. In this regard, the number 40 was associated with the notion of the end of the account, actually taking the meaning of "many", which is widely reflected in the Russian language and Russian literature (see 40 (number) , Sorokovina , Forty-forty ) ^[13] . Some ancient ideas about the number 41 as unfavorable are connected with this.

• According to Siberian belief, if a hunter killed forty bears, then he himself died from the forty-first ^[13] (according to another version, the fortieth bear ^{[14] [15]} becomes fatal). With this, in particular, are connected the lines of the poet Nekrasov :

After all, he hit a creeping thing!
Was my son not removed?
Forty bears faked on a spear -
At the forty-first sploshal!

("In the village")

Other mentions: first, song ; the story "Forty-first" on proza.ru/2013/02/23/1422.

Chuvashi

In Chuvash 41 - one of the sacred numbers, used in some ceremonies ^[16] .

• At 41, Publius Elijah Adrian became Roman emperor, succeeding Emperor Trajan ( 117 AD ).
• 41 states included the German Union in 1815 (the year the union was formed).
• The 41st km of the Leningradskoye Highway ( Russia , Moscow region ) is the closest to Moscow exactly the confirmed place of the offensive of the Wehrmacht troops in 1941 (the Soviet offensive on the Leningrad direction began from this line) ^[17] .
• During World War II, the following military formations with the sequence number 41: 41st Fighter Aviation Regiment , 41st Motorized Corps , 41st Separate Ski Battalion , 41st Separate Motorized Pontoon-Bridge Battalion , 41st Separate Tank Battalion , 41st Infantry Corps , 41st Tank Corps , 41st Air Division , 41st Army , 41st Infantry Division (1st formation) , 41st Infantry Division (2nd formation) , 41- I am a cavalry division , 41st tank division .
• “41 for Freedom” is the unofficial name of the group of strategic nuclear submarines Polaris, built by the United States in 1958-1967. and later replaced. The boats were named after famous figures of the United States. This number was limited to the 1972 Soviet-American SALT-1 treaty . These boats became the main enemy of the USSR during the Cold War .
• George Bush is the 41st president of the United States .

Olympics

• 41 disciplines are included in the summer Olympic sports
• The 41st General Assembly of the European Olympic Committee decided to hold the first ever European Games.

Bandy

• Dinamo (Moscow) scored 41 unanswered goals against Lokomotiv (Orenburg) in the Russian hockey championship on December 28, 2006 . The match ended with a score of 41: 0.

Russian literature

• "Forty-first" - the story of the writer Boris Lavrenyov (1924, filmed in 1927 and 1956 ) about the love of the red arrow Maryuki and the White Guard lieutenant during the Civil War ^[18]
• the story "№ 41" ^[19]

Mari Literature

• Forty-one foal , Mari tale
• Forty-one fiction , Mari tale
• Boy hero , Mari fairy tale

Literature of other nations

• About forty bald and one oblique Crimean Tatar fairy tale
• 41st sonnet of Shakespeare ^[20]

Films:

• Forty-first (film, 1927) , Forty-first (film, 1956) - films based on the book by Boris Lavrenyov
• In June of the 41st (Russia, 2008), Spring of forty-first (Poland, 2008), Forty-first year: The Opposition , Apples of the forty-first year , as well as many documentaries - about the Second World War (year 1941).
• 0-41 * is an Indian documentary drama film directed by Senna Hegde .

• 41 , 141 , 241 , 341 , 441 , 541 , 641 , 741 , 841 , 941 , 1041 , 1141 , 1241 , 1341 , 1441 , 1541 , 1641 , 1741 , 1841 , 1941 , 2041
• 41 BC er
• The 41st day of the year - February 10

• ASCII code of the symbol " ) "
• Area code Switzerland +41
• Montana - 41st US State
• Loir et Cher - 41st Department in France
• In Russia, 41 is the code of the Kamchatka Territory
• 41 Channel Home
• Sum 41 - Canadian punk band
• Process C-41 - Film Processing
• "Forty-one" - Zelenograd district newspaper
• "Forty-one" - a card game
• 41 ° ("forty first degree") - a group of poets in Tiflis at the beginning of the twentieth century
• exactly 41 letters are present in the alphabet of the Tofalar language , in the Cyrillic version of the Karakalpak alphabet
• Roads P41 (in Belarus), M41 (in Central Asia), Highway 41 (in Israel)
• Geography: 41st east longitude (in Danish: 41. østlige længdekreds ), 41st west longitude
• Railway: 41 km (platform) , Area 41 km
• Symphony number 41 (Mozart)