Misplaced Pages

#376623

55-475: The current century is the 21st century AD, under the Gregorian calendar . Twenty-one is the fifth distinct semiprime , and the second of the form 3 × q {\displaystyle 3\times q} where q {\displaystyle q} is a higher prime. It is a repdigit in quaternary (111 4 ). As a biprime with proper divisors 1 , 3 and 7 , twenty-one has

110-2382: A {\displaystyle a} and n − a {\displaystyle n-a} only has factors 2 {\displaystyle 2} and 5 {\displaystyle 5} (for a representation in base ten ). Let A ( n ) {\displaystyle A(n)} denote the quantity of the numbers smaller than n {\displaystyle n} that only have factor 2 {\displaystyle 2} and 5 {\displaystyle 5} and that are coprime to n {\displaystyle n} , we instantly have φ ( n ) 2 < A ( n ) {\displaystyle {\frac {\varphi (n)}{2}}<A(n)} . We can easily see that for sufficiently large n {\displaystyle n} , A ( n ) ∼ log 2 ⁡ ( n ) log 5 ⁡ ( n ) 2 = ln 2 ⁡ ( n ) 2 ln ⁡ ( 2 ) ln ⁡ ( 5 ) . {\displaystyle A(n)\sim {\frac {\log _{2}(n)\log _{5}(n)}{2}}={\frac {\ln ^{2}(n)}{2\ln(2)\ln(5)}}.} However, φ ( n ) ∼ n e γ ln ⁡ ln ⁡ n {\displaystyle \varphi (n)\sim {\frac {n}{e^{\gamma }\;\ln \ln n}}} where A ( n ) = o ( φ ( n ) ) {\displaystyle A(n)=o(\varphi (n))} as n {\displaystyle n} approaches infinity ; thus φ ( n ) 2 < A ( n ) {\displaystyle {\frac {\varphi (n)}{2}}<A(n)} fails to hold for sufficiently large n {\displaystyle n} . In fact, for every n > 2 {\displaystyle n>2} , we have So φ ( n ) 2 < A ( n ) {\displaystyle {\frac {\varphi (n)}{2}}<A(n)} fails to hold when n > 273 {\displaystyle n>273} (actually, when n > 33 {\displaystyle n>33} ). Just check

165-447: A i {\displaystyle a_{i}} must be either zero or a positive integer up to ⁠ n − 1 {\displaystyle n-1} ⁠ .) X can be expressed as X is a harshad number in base n if: A number which is a harshad number in every number base is called an all-harshad number , or an all-Niven number . There are only four all-harshad numbers: 1 , 2 , 4 , and 6 . The number 12

220-405: A b {\displaystyle {\tfrac {a}{b}}} and b a {\displaystyle {\tfrac {b}{a}}} is a terminating decimal; see proof below: For any a {\displaystyle a} coprime to n {\displaystyle n} and n − a {\displaystyle n-a} , the condition above holds when one of

275-444: A few numbers to see that the complete sequence of numbers having this property is { 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 11 , 12 , 15 , 21 } . {\displaystyle \{2,3,4,5,6,7,8,9,11,12,15,21\}.} 21 is the smallest natural number that is not close to a power of two ( 2 n ) {\displaystyle (2^{n})} , where

330-564: A given number base is an integer t such that there exists some harshad number N whose digit sum is t , and t , written in that base, terminates N written in the same base. For example, 18 is a Nivenmorphic number for base 10: Sandro Boscaro determined that for base 10 all positive integers are Nivenmorphic numbers except 11 . In fact, for an even integer n > 1, all positive integers except n +1 are Nivenmorphic numbers for base n , and for an odd integer n > 1, all positive integers are Nivenmorphic numbers for base n . e.g.

385-634: A mathematician from India . The word "harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver. The term "Niven number" arose from a paper delivered by Ivan M. Niven at a conference on number theory in 1977. Stated mathematically, let X be a positive integer with m digits when written in base n , and let the digits be a i {\displaystyle a_{i}} ( i = 0 , 1 , … , m − 1 {\displaystyle i=0,1,\ldots ,m-1} ). (It follows that

440-552: A new biological era for the human species. Anxiety and depression rates have risen in the United States and many other parts of the world. However, suicide rates have fallen in Europe and most of the rest of the world so far this century, declining 29% globally between 2000 and 2018, despite rising 18% in the United States in the same period. The decline in suicide has been most notable among Chinese and Indian women,

495-480: A number that is more than 1, but less than the prime, and will not be divisible. For example: 11 is not harshad in base 10 because the sum of its digits “11” is 1 + 1 = 2, and 11 is not divisible by 2; while in base 12 the number 11 may be represented as “B”, the sum of whose digits is also B. Since B is divisible by itself, it is harshad in base 12. Although the sequence of factorials starts with harshad numbers in base 10, not all factorials are harshad numbers. 432!

550-449: A prime aliquot sum of 11 within an aliquot sequence containing only one composite number (21, 11 , 1 , 0 ); it is the second composite number with an aliquot sum of 11 , following 18 . 21 is the first member of the second cluster of consecutive discrete semiprimes (21, 22 ), where the next such cluster is ( 33 , 34 , 35 ). There are 21 prime numbers with 2 digits. There are A total of 21 prime numbers between 100 and 200 . 21

605-557: A robotic spacecraft on the far side of the Moon , the first to do so. On 23 August 2023, with the Chandrayaan-3 Mission , India became the first country to touch down near the lunar south pole . War and violence have declined considerably compared to the 20th century, continuing the post-World War II trend called Long Peace . However, since the beginning of the 2020s, geopolitical tensions and wars have been rising across

SECTION 10

#1732765179377

660-601: A serious concern; UN Chief António Guterres , for instance, has described it as an " existential threat " to humanity. Furthermore, the Holocene extinction event, the sixth most significant extinction event in the Earth's history, continues with the widespread degradation of highly biodiverse habitats as a by-product of human activity. Economically and politically, the United States and Western Europe were dominant at

715-427: A sum of 427 when excluding a square of side length 7 {\displaystyle 7} ; this sum represents the largest square-free integer over a quadratic field of class number two, where 163 is the largest such ( Heegner ) number of class one. 427 number is also the first number to hold a sum-of-divisors in equivalence with the third perfect number and thirty-first triangular number ( 496 ), where it

770-697: Is a base-12 harshad number are (written in base 10): Smallest k such that k ⋅ n {\displaystyle k\cdot n} is not a base-12 harshad number are (written in base 10): Similar to base 10, not all factorials are harshad numbers in base 12. After 7! (= 5040 = 2B00 in base 12, with digit sum 13 in base 12, and 13 does not divide 7!), 1276! is the next that is not. (1276! has digit sum 14201 = 11 × 1291 in base 12, thus does not divide 1276!) Cooper and Kennedy proved in 1993 that no 21 consecutive integers are all harshad numbers in base 10. They also constructed infinitely many 20-tuples of consecutive integers that are all 10-harshad numbers,

825-429: Is a harshad number in all bases except octal . Given the divisibility test for 9, one might be tempted to generalize that all numbers divisible by 9 are also harshad numbers. But for the purpose of determining the harshadness of n , the digits of n can only be added up once and n must be divisible by that sum; otherwise, it is not a harshad number. For example, 99 is not a harshad number, since 9 + 9 = 18, and 99

880-771: Is also the fiftieth number to return 0 {\displaystyle 0} in the Mertens function . While the twenty-first prime number 73 is the largest member of Bhargava's definite quadratic 17– integer matrix Φ s ( P ) {\displaystyle \Phi _{s}(P)} representative of all prime numbers, Φ s ( P ) = { 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , 47 , 67 , 73 } , {\displaystyle \Phi _{s}(P)=\{2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,67,{\mathbf {73}}\},}

935-423: Is also the first non-trivial octagonal number . It is the fifth Motzkin number , and the seventeenth Padovan number (preceded by the terms 9 , 12 , and 16 , where it is the sum of the first two of these). In decimal , the number of two-digit prime numbers is twenty-one (a base in which 21 is the fourteenth Harshad number ). It is the smallest non-trivial example in base ten of a Fibonacci number (where 21

990-408: Is not divisible by 18. The base number (and furthermore, its powers) will always be a harshad number in its own base, since it will be represented as "10" and 1 + 0 = 1. All numbers whose base b digit sum divides b −1 are harshad numbers in base b . For a prime number to also be a harshad number it must be less than or equal to the base number, otherwise the digits of the prime will add up to

1045-470: Is the 8th member, as the sum of the preceding terms in the sequence 8 and 13 ) whose digits ( 2 , 1 ) are Fibonacci numbers and whose digit sum is also a Fibonacci number ( 3 ). It is also the largest positive integer n {\displaystyle n} in decimal such that for any positive integers a , b {\displaystyle a,b} where a + b = n {\displaystyle a+b=n} , at least one of

1100-682: Is the base, k is a relatively large power, and N is a constant. Given one such suitably chosen sequence, we can convert it to a larger one as follows: Thus our initial sequence yields an infinite set of solutions. The smallest naturals starting runs of exactly n consecutive 10-harshad numbers (i.e., the smallest x such that x , x + 1 , ⋯ , x + n − 1 {\displaystyle x,x+1,\cdots ,x+n-1} are harshad numbers but x − 1 {\displaystyle x-1} and x + n {\displaystyle x+n} are not) are as follows (sequence A060159 in

1155-474: Is the current century in the Anno Domini or Common Era , in accordance with the Gregorian calendar . It began on 1 January 2001 and will end on 31 December 2100 . It is the first century of the 3rd millennium . The rise of a global economy and Third World consumerism marked the beginning of the century, along with increased private enterprise and deepening concern over terrorism after

SECTION 20

#1732765179377

1210-615: Is the first Blum integer , since it is a semiprime with both its prime factors being Gaussian primes . While 21 is the sixth triangular number , it is also the sum of the divisors of the first five positive integers : 1 + 2 + 3 + 4 + 5 + 6 = 21 1 + ( 1 + 2 ) + ( 1 + 3 ) + ( 1 + 2 + 4 ) + ( 1 + 5 ) = 21 {\displaystyle {\begin{aligned}1&+2+3+4+5+6=21\\1&+(1+2)+(1+3)+(1+2+4)+(1+5)=21\\\end{aligned}}} 21

1265-527: Is the first that is not. (432! has digit sum 3897 = 3 × 433 in base 10, thus not dividing 432!) The smallest k such that k ⋅ n {\displaystyle k\cdot n} is a harshad number are The smallest k such that k ⋅ n {\displaystyle k\cdot n} is not a harshad number are The harshad numbers in base 12 are: where A represents ten and B represents eleven. Smallest k such that k ⋅ n {\displaystyle k\cdot n}

1320-419: Is the sum of at most k {\displaystyle k} harshad numbers, that is, the set of harshad numbers is an additive basis . The number of ways that each natural number 1, 2, 3, ... can be written as sum of two harshad numbers is: The smallest number that can be written in exactly 1, 2, 3, ... ways as the sum of two harshad numbers is: A Nivenmorphic number or harshadmorphic number for

1375-479: The o ( 1 ) {\displaystyle o(1)} term uses Big O notation . Every natural number not exceeding one billion is either a harshad number or the sum of two harshad numbers. Conditional to a technical hypothesis on the zeros of certain Dedekind zeta functions , Sanna proved that there exists a positive integer k {\displaystyle k} such that every natural number

1430-570: The COVID-19 pandemic began to rapidly spread worldwide, causing more than 7 million reported deaths, and around 18.2 to 33.5 million estimated deaths, while at the same time, causing severe global economic disruption , including the largest global recession since the Great Depression in the 1930s. Due to the sudden proliferation of internet-accessible mobile devices , such as smartphones becoming ubiquitous worldwide beginning in

1485-598: The COVID-19 pandemic . 2000s 2010s 2020s Association football is the most popular sport worldwide with the FIFA World Cup being the most viewed football event. Other sports such as rugby, cricket, baseball, basketball, ice hockey, tennis, and golf are popular globally. In cricket , the emergence of the Twenty20 format and the creation of the Indian Premier League led to changes in

1540-805: The Federal Republic of Yugoslavia . The Islamic Republic of Afghanistan replaced the Transitional Islamic State of Afghanistan The Federal Democratic Republic of Nepal replaced the Kingdom of Nepal . The National Transitional Council of Libya replaced the Great Socialist People's Libyan Arab Jamahiriya . The State of Libya replaced the National Transitional Council of Libya . The Islamic Emirate of Afghanistan replaced

1595-482: The Internet . Telecommunications in the early 21st century are much more advanced and universal than they were in the late 20th century. Only a few percent of the world's population were Internet users and cellular phone owners in the late 1990s; while as of 2023, 67% of the world's population is online, and 78% of all people aged 10 and above own a mobile phone. In the 2010s, artificial intelligence , mainly in

1650-574: The Islamic Republic of Afghanistan . These territories have declared independence and secured relative autonomy but they have only been recognized by some UN member states: These territories have declared independence and secured relative autonomy but they have been recognized by no one: These territories were annexed from a sovereign country, the action has only been recognized by some UN member states: These territories were ceded to another country: The Digital Revolution continued into

1705-750: The OEIS ): By the previous section, no such x exists for n > 20. {\displaystyle n>20.} If we let N ( x ) {\displaystyle N(x)} denote the number of harshad numbers ≤ x {\displaystyle \leq x} , then for any given ε > 0 , {\displaystyle \varepsilon >0,} as shown by Jean-Marie De Koninck and Nicolas Doyon; furthermore, De Koninck, Doyon and Kátai proved that where c = ( 14 / 27 ) log ⁡ 10 ≈ 1.1939 {\displaystyle c=(14/27)\log 10\approx 1.1939} and

21 (number) - Misplaced Pages Continue

1760-761: The Russian Invasion of Ukraine and the Israel-Hamas war . The United States has remained the sole global superpower , while China is now considered to be an emerging superpower . In 2022, 45% of the world's population lived in "some form of democracy", although only 8% lived in "full democracies". The United Nations estimates that by 2050, two thirds of the world's population will be urbanized . The world economy expanded at high rates from $ 42 trillion in 2000 to $ 101 trillion in 2022, and though many economies rose at greater levels, some gradually contracted. The European Union greatly expanded in

1815-712: The September 11 attacks in 2001. The NATO intervention in Afghanistan and the United States -led coalition intervention in Iraq in the early 2000s, as well as the overthrow of several regimes during the Arab Spring in the early 2010s, led to mixed outcomes in the Arab world , resulting in several civil wars and political instability. The 2020s saw an increase in wars across the world, as seen with conflicts such as

1870-549: The 21st century, adding 13 member states , but the United Kingdom withdrew . Most EU member states introduced a common currency, the Euro . The North Atlantic Treaty Organization (NATO), also greatly expanded, adding 13 member states . Effects of global warming and rising sea levels exacerbated the ecological crises , with eight islands disappearing between 2007 and 2014. Globally, from January 2020 to May 2023,

1925-770: The 21st century. This is a list of sovereign states that have gained independence in the 21st century and have been recognized by the UN . These nations gained sovereignty through government reform. The Union of the Comoros replaced the Federal Islamic Republic of the Comoros The Transitional Islamic State of Afghanistan replaced the Islamic State of Afghanistan . The State Union of Serbia and Montenegro replaced

1980-470: The Nivenmorphic numbers in base 12 are OEIS :  A011760 (all positive integers except 13). The smallest number with base 10 digit sum n and terminates n written in base 10 are: (0 if no such number exists) Bloem (2005) defines a multiple harshad number as a harshad number that, when divided by the sum of its digits, produces another harshad number. He states that 6804 is "MHN-4" on

2035-432: The beginning of the century; by the 2010s , China became an emerging global superpower and, by some measures, the world's largest economy . In terms of purchasing power parity , India's economy became more significant than Japan 's around 2011. Bitcoin and other cryptocurrencies are decentralized currencies that are not controlled by any central bank. These currencies are increasing in popularity worldwide due to

2090-657: The early 2010s, more than two thirds of the world's population obtained access to the Internet by 2023. After the success of the Human Genome Project , DNA sequencing services became available and affordable. There were significant improvements in the complexity of artificial intelligence , with American companies, universities, and research labs pioneering advances in the field. Generative AI -based applications such as ChatGPT and DALL-E have accumulated billions of users, and allow users to instantly generate complex texts, images, art, and video, comparable to

2145-741: The early 21st century with mobile phone usage and Global Internet usage growing massively, becoming available to many more people, with more applications and faster speeds. Social networking emerged in the mid-2000s as a popular form of social communication, partly replacing much of the function of email , message boards and instant messaging services. Twitter , Facebook , YouTube , Instagram , Snapchat and WeChat are all major examples of social media which have gained widespread popularity. The use of webcams and front-facing cameras on PCs and related devices, and services such as Skype , Zoom and FaceTime , have made video calling and video conferencing widespread. Their use hugely increased during

2200-483: The early years of previous centuries were commonly pronounced as, for example, "eighteen oh five" (for 1805) and "nineteen oh five" (for 1905). Generally, the early years of the 21st century were pronounced as in "two-thousand (and) five," with a change taking place around 2010, when pronunciations often shifted between the early-years form of "two-thousand and ten" and the traditionally more concise form of "twenty-ten." The Vancouver Olympics , which took place in 2010,

2255-533: The elderly, and middle-aged Russian men. The entire written works of humanity, from the beginning of recorded history to 2003, in all known languages , are estimated to amount to five exabytes of data. Since 2003, with the beginning of social media and " user-generated content ", the same amount of data is created every two days. With the AI boom of the 2020s gaining international prominence, as of 2024, mass-produced AI-generated content comprised over half of

21 (number) - Misplaced Pages Continue

2310-445: The expanding availability of the internet and are mainly used as a store of value . There is an ongoing impact of technological unemployment due to automation and computerization : the rate at which jobs are disappearing—due to machines replacing them—is expected to escalate. Automation alters the number of jobs and the skills demands of industries. As of 2019, the production output of first world nations ' manufacturing sectors

2365-621: The form of deep learning and machine learning , became more prevalent and in the early 2020s, with the rise of generative AI , the AI boom began. As of 2022, 8.6% of the world's population still lacked access to electricity. In 2001, Dennis Tito became the first space tourist , beginning the era of commercial spaceflight . Entrepreneurs Elon Musk and Richard Branson are working towards commercial space exploration, colonization and tourism, while China and India have made substantial strides in their space programs. On 3 January 2019, China landed

2420-466: The general consensus is that there are between 6,000 and 7,000 languages currently spoken. Between 50% and 90% of those will have become extinct by the year 2100. The top 20 languages spoken by more than 50 million speakers each, are spoken by 50% of the world's population. In contrast, many of the other languages are spoken by small communities, most of them with fewer than 10,000 speakers. Some territories and states have gained independence during

2475-465: The nature of the sport. American swimmer Michael Phelps won an Olympic record setting 8 Gold medals at the 2008 Summer Olympics . Harshad number In mathematics , a harshad number (or Niven number ) in a given number base is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers in base n are also known as n -harshad (or n -Niven ) numbers. Harshad numbers were defined by D. R. Kaprekar ,

2530-508: The possible adverse effects of social media in influencing citizen's views, particularly regarding the 2016 United States presidential election . The world population was about 6.1 billion at the start of the 21st century and reached 8 billion by November 2022. It is estimated to reach nearly 8.6 billion by 2030, and 9.8 billion by 2050. According to the United Nations World Urbanization prospects, 60% of

2585-590: The range of nearness is ± n . {\displaystyle \pm {n}.} Twenty-one is the smallest number of differently sized squares needed to square the square . The lengths of sides of these squares are { 2 , 4 , 6 , 7 , 8 , 9 , 11 , 15 , 16 , 17 , 18 , 19 , 24 , 25 , 27 , 29 , 33 , 35 , 37 , 42 , 50 } {\displaystyle \{2,4,6,7,8,9,11,15,16,17,18,19,24,25,27,29,33,35,37,42,50\}} which generate

2640-450: The retirement age has been called for in view of an increase in life expectancy and has been put in place in many jurisdictions. As of 2009, Ethnologue catalogued 6,909 living human languages. The exact number of known living languages will vary from 5,000 to 10,000, generally depending on the precision of one's definition of "language", and in particular, on how one classifies dialects . Estimates vary depending on many factors, but

2695-699: The smallest of which exceeds 10 . H. G. Grundman  ( 1994 ) extended the Cooper and Kennedy result to show that there are 2 b but not 2 b + 1 consecutive b -harshad numbers for any base b . This result was strengthened to show that there are infinitely many runs of 2 b consecutive b -harshad numbers for b = 2 or 3 by T. Cai  ( 1996 ) and for arbitrary b by Brad Wilson in 1997. In binary , there are thus infinitely many runs of four consecutive harshad numbers and in ternary infinitely many runs of six. In general, such maximal sequences run from N · b − b to N · b + ( b − 1), where b

2750-546: The sophistication of human work. Research into outer space greatly accelerated in the 2020s, with the United States mainly dominating space exploration , including the James Webb Space Telescope , Ingenuity helicopter , Lunar Gateway , and Artemis program . There is a lack of general agreement over how to pronounce specific years of the 21st century in English. Academics have pointed out that

2805-490: The twenty-first composite number 33 is the largest member of a like definite quadratic 7–integer matrix Φ s ( 2 Z ≥ 0 + 1 ) = { 1 , 3 , 5 , 7 , 11 , 15 , 33 } {\displaystyle \Phi _{s}(2\mathbb {Z} _{\geq 0}+1)=\{1,3,5,7,11,15,{\mathbf {33}}\}} representative of all odd numbers. 21 is: 21st century The 21st century

SECTION 50

#1732765179377

2860-480: The world's human population are projected to live in megacities and megalopolises by 2030, 70% by 2050, and 90% by 2080. Life expectancy has increased as child mortality continues to decline. A baby born in 2019, for example, will, on average (globally), live to 73 years—27 years longer than the global average of someone born in 1950. Ten million Britons (16% of the United Kingdom population) will, on average, live to 100 or older. Climate change remains

2915-453: The world, as seen with the deterioration of US-China relations , Russian Invasion of Ukraine , Tigray War , Sudanese civil war , Israel–Hamas War , etc. Poverty is still widespread globally, but fewer people live in the most extreme forms of poverty. In 1990, 37.9% of the world's population lived in extreme poverty ; by 2022, this had dropped to just 9%. The Facebook–Cambridge Analytica data scandal drew international attention to

2970-755: Was being officially referred to by Vancouver 2010 as "the twenty-ten Olympics". Technologies such as ultrasound , prenatal genetic testing and genetic engineering have advanced significantly. Because of sex-selective abortion , fewer girls have been born in the 21st century (and since the early 1980s) compared to past centuries, mostly because of son preference in East and South Asia. In 2014, only 47 percent of Indian births were of girls. This has led to an increase in bachelors in countries such as China and India. The first genetically modified children were born November 2018 in China to significant controversy, beginning

3025-452: Was doubled when compared to 1984 output; but it is now produced with one-third fewer workers and at significantly reduced operating costs . Half of all jobs with requirements lower than a bachelor's degree are currently in the process of being replaced with partial- or full-automation. The World Economic Forum forecast that 65% of children entering primary school will end up in jobs or careers that currently do not yet exist. A rise in

#376623