19 (பத்தொன்பது) (nineteen) என்பதுஇயல் எண் ஆகும். இது 18 இன் தொடரி மற்றும் 20 இன் முன்னி ஆகும்.இது பகா எண் ஆகும்.

கணிதத்தில்

பத்தொன்பது என்பது எட்டாவது பகா எண் ஆகும்.

எண் கோட்பாடு

[[இரட்டைப் பகாத்தனி |இரட்டை பகா எண்]](twin prime) என்பது இரண்டு பகா எண்கள் (பகாத்தனிகள்) தங்களுக்கு இடையே உள்ள வேறுபாடு இரண்டு ஆக உள்ளவை (p, p+2). ஆகும்.எடுத்துக் காட்டாக இரட்டை பகாத்தனி எண்களாகிய (5, 7) ஐக் கூறலாம். 19 ஆனது 17 உடன் இரட்டை பகா எண்ணை உருவாக்குகிறது.,^[1] அதே போன்று 19 ஆனது 23 உடன் நான்ககல் பகாத்தனியை (cousin prime) உருவாக்குகிறது. ^[2] மேலும் ஆறகல் பகாத்தனியை எண் 13 உடன் உருவாக்குகிறது.^[3] 19 ஆனது ஐந்தாவது மத்திய மூவுறுப்புக்கோவையின் குணகமாகும். (Central trinomial coefficient ),^[4] வாரிங் தேற்றமானது ஒவ்வொரு முழு எண்ணும் 19க்கு மிகையாகாத எண்ணிக்கையில் அத்தனை நாற்படியங்களின் கூட்டுத்தொகையாகும்.(பார்க்க, வாரிங் தேற்றம்).^[5] It is the number of compositions of 8 into distinct parts.^[6]

19 is the eighth strictly non-palindromic number in any base, following 11 and preceding 47.^[7].19 ஆனது 6 க்கு அடுத்த இரண்டாவது எண்முக எண்ணாகும் மற்றும் ஆறாவது ஹீக்னர் எண்ணாகும்.

^[8]

In the Engel expansion of பை (கணித மாறிலி),^[9] 19 is the seventh term following ${1, 1, 1, 8, 8, 17}$ and preceding ${300, 1991, ...}$ . The sum of the first terms preceding 17 is in equivalence with 19, where its prime index (8) are the two previous members in the sequence.

பகா எண்ணின் பண்புகள்

19 ஆனது ஏழாவது மெர்சன் பகாத்தணி அடுக்காகும்.^[10] It is the second Keith number, and more specifically the first Keith prime.^[11] In decimal, 19 is the third full reptend prime,^[12] and the first prime number that is not a permutable prime, as its reverse (91) is composite (where 91 is also the fourth centered nonagonal number).^[13]

19 × 91 = 1729, the first Hardy-Ramanujan number or taxicab number, also a Harshad number in base-ten, as it's divisible by the sum of its digits, 19.^[14]^[15]

1729 is also the nineteenth dodecagonal number.^[16]

19, alongside 109, 1009, and 10009, are all prime (with 109 also full reptend), and form part of a sequence of numbers where inserting a digit inside the previous term produces the next smallest prime possible, up to scale, with the composite number 9 as root.^[17] 100019 is the next such smallest prime number, by the insertion of a 1.

Numbers of the form 10_n9 equivalent to 10^x + 9 with x = n + 1, where n is the number of zeros in the term, are prime for $n = {0, 1, 2, 3, 8, 17, 21, 44, 48, 55, 68, 145, 201, 271, 2731, 4563}$ , and probably prime for $n = {31811, 43187, 48109, 92691}$ .^[18]

Otherwise, $R_{19}$ is the second base-10 repunit prime, short for the number $1111111111111111111$ .^[19]

The sum of the squares of the first nineteen primes is divisible by 19.^[20]

Figurate numbers and magic figures

19 is the third centered triangular number as well as the third centered hexagonal number.^[21]^[22]

The 19th triangular number is 190, equivalently the sum of the first 19 non-zero integers, that is also the sixth centered nonagonal number.^[23]^[13]

19 is the first number in an infinite sequence of numbers in decimal whose digits start with 1 and have trailing 9's, that form triangular numbers containing trailing zeroes in proportion to 9s present in the original number; i.e. 19900 is the 199th triangular number, and 1999000 is the 1999th.^[24]

Like 19, 199 and 1999 are also both prime, as are 199999 and 19999999. In fact, a number of the form 19_n, where n is the number of nines that terminate in the number, is prime for:

n = {1, 2, 3, 5, 7, 26, 27, 53, 147, 236, 248, 386, 401}

.^[25]

The number of nodes in regular hexagon with all diagonals drawn is nineteen.^[26]

Distinguishably, the only nontrivial normal magic hexagon is composed of nineteen cells, where every diagonal of consecutive hexagons has sums equal to 38, or twice 19.^[27]
A hexaflexagon is a strip of nineteen alternating triangular faces that can flex into a regular hexagon, such that any two of six colorings on triangles can be oriented to align on opposite sides of the folded figure.^[28]
Nineteen is also the number of one-sided hexiamonds, meaning there are nineteen ways of arranging six equiangular triangular polyforms edge-to-edge on the plane without turn-overs (and where holes are allowed).^[29]

${\tfrac {1}{19}}$ can be used to generate the first full, non-normal prime reciprocal magic square in decimal whose rows, columns and diagonals — in a 18 x 18 array — all generate a magic constant of 81 = 9².^[30]

The next prime number to generate a like-magic square in base-ten is 383,^[31] the seventy-sixth prime number (where 19 × 4 = 76).^[32] A regular 19 x 19 magic square, on the other hand, has a magic constant $M_{19}$ of 3439 = 19 × 181.^[33]

Collatz problem

The Collatz sequence for nine requires nineteen steps to return to one, more than any other number below it.^[34] On the other hand, nineteen requires twenty steps, like eighteen. Less than ten thousand, only thirty-one other numbers require nineteen steps to return to one:

{56, 58, 60, 61, 352, 360, 362, 368, 369, 372, 373, 401, 402, 403, 2176, ..., and 2421}

.^[35]

In abstract algebra

The projective special linear group $\mathrm {L(19)}$ represents the abstract structure of the 57-cell: a universal 4-polytope with a total of one hundred and seventy-one (171 = 9 × 19) edges and vertices, and fifty-seven (57 = 3 × 19) hemi-icosahedral cells that are self-dual.^[36]

In total, there are nineteen Coxeter groups of non-prismatic uniform honeycombs in the fourth dimension: five Coxeter honeycomb groups exist in Euclidean space, while the other fourteen Coxeter groups are compact and paracompact hyperbolic honeycomb groups.

There are also specifically nineteen uniform honeycombs inside the Euclidean ${\tilde {C}}_{4}$ tesseractic honeycomb group in 4-space. In 5-space, there are nineteen uniform polytopes with $\mathrm {A} _{5}$ simplex symmetry.

There are infinitely many finite-volume Vinberg polytopes up through dimension nineteen, which generate hyperbolic tilings with degenerate simplex quadrilateral pyramidal domains, as well as prismatic domains and otherwise.^[37]

Vinberg polytopes of lowest rank $n + 2$ mirrors exist up through the seventeenth dimension, where there exists a unique figure with nineteen facets.^[38] It is expressed with the simple Dynkin diagram, .

On the other hand, a cubic surface is the zero set in $\mathbb {P^{3}}$ of a homogeneous cubic polynomial in four variables $f=c_{3}000x_{1}^{3}+c_{2}100x_{1}^{2}x_{2}+c_{1}200x^{1}x_{2}^{2}+c_{0}300x_{2}^{3}+\cdots +c_{0}003x_{4}^{3},$ a polynomial with a total of twenty coefficients, which specifies a space for cubic surfaces that is 19-dimensional.^[39]

Finite simple groups

19 is the eighth consecutive supersingular prime. It is the middle indexed member in the sequence of fifteen such primes that divide the order of the Friendly Giant $\mathrm {F_{1}}$ , the largest sporadic group: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71}.^[40]

Janko groups $\mathrm {J_{1}}$ and $\mathrm {J_{3}}$ are the two-smallest of six pariah groups that are not subquotients of $\mathrm {F_{1}}$ , which contain 19 as the largest prime number that divides their orders.^[41]

\mathrm {J_{1}}

holds

(2,3,7)

as standard generators

(a, b, ab)

that yield a semi-presentation where

o (abab 2) = 19

, while

\mathrm {J_{3}}

holds as standard generators

(2A, 3A, 19)

, where

o ([a, b]) = 9

.^[42]^[43]

$10944$ is the dimensionality of the minimal faithful complex representation of O'Nan group $O'N$ — the second-largest after $\mathrm {F_{1}}$ of like-representation in ${\text{dim}}{\text{ }}196,883$ and largest amongst the six pariahs^[44] — whose value lies midway between primes (10939, 10949), the latter with a prime index of $1330$ ,^[45] which is the nineteenth tetrahedral number.^[46]
On the other hand, the Tits group $\mathrm {T}$ , as the only non-strict group of Lie type that can loosely categorize as sporadic, has group order $211 \cdot 3 3 \cdot 5 2 \cdot 13$ , whose prime factors (inclusive of powers) generate a sum equal to 54, which is the smallest non-trivial 19-gonal number.^[47]

In the Happy Family of sporadic groups, nineteen of twenty-six such groups are subquotients of the Friendly Giant, which is also its own subquotient.^[48] If the Tits group is indeed included as a group of Lie type,^[49] then there are nineteen classes of finite simple groups that are not sporadic groups.

Heegner number

19 is the sixth Heegner number.^[50] 67 and 163, respectively the 19th and 38th prime numbers, are the two largest Heegner numbers, of nine total.

The sum of the first six Heegner numbers 1, 2, 3, 7, 11, and 19 sum to the seventh member and fourteenth prime number, 43. All of these numbers are prime, aside from the unit. In particular, 163 is relevant in moonshine theory.

Science

The James Webb Space Telescope's optical elements are in an array of 19 hexagons, wherein 18 segmented primary mirrors focus light into a centered secondary mirror situated above the collecting mirrors, that in-turn reflects back onto the telescope's central imagers. It is a shape akin to the order-3 magic hexagon.

Religion

Islam

A group called United Submitter International claim the Quran has mathematical structure based on the number 19. The gematrical value of WAHD = 6+1+8+4=19, Wahd means 'One' (God) to the first verse (1:1), known as Bas-malah, consists of 19 Arabic letters or the Quran consists of 114 (19x6) surat etc.^{[சான்று தேவை]}

Baháʼí Faith

In the Bábí and Baháʼí Faiths, a group of 19 is called a Váhid, a Unity (அரபி: واحد, romanized: wāhid, lit. 'one'). The numerical value of this word in the Abjad numeral system is 19.

The Baháʼí calendar is structured such that a year contains 19 months of 19 days each (along with the intercalary period of Ayyám-i-Há), as well as a 19-year cycle and a 361-year (19x19) supercycle.
The Báb and his disciples formed a group of 19.
There were 19 Apostles of Bahá'u'lláh.

Celtic paganism

19 is a sacred number of the goddess Brigid because it is said to represent the 19-year cycle of the Great Celtic Year and the amount of time it takes the Moon to coincide with the winter solstice.^[51]

Music

Nineteen has been used as an alternative to twelve for a division of the octave into equal parts. This idea goes back to Salinas in the sixteenth century, and is interesting in part because it gives a system of meantone tuning, being close to 1/3 comma meantone. See 19 equal temperament.
Some organs use the 19th harmonic to approximate a minor third.^{[சான்று தேவை]}

Literature

Stephen King's eight-book epic The Dark Tower saga employs the number 19 in books The Dark Tower: The Gunslinger, The Dark Tower V: Wolves of the Calla, The Dark Tower VI: Song of Susannah, The Dark Tower VII: The Dark Tower, and The Wind Through the Keyhole as a mysterious and important number. They refer to the "ka-tet of 19", "Directive Nineteen," many names add up to 19, 19 seems to permeate every aspect of Roland and his traveler's lives. In addition, the number ends up being a powerful key.

Games

Though the maximum score for a cribbage hand is 29, there is no combination of cards that adds up to 19 points. Many cribbage players, therefore, jokingly refer to a zero-point hand as "a 19 hand".

Age 19

In four countries, 19 is the age of majority.
In two countries, 19 is the minimum legal drinking age.
In three countries, 19 is the minimum legal smoking age.
The minimum age for the NBA draft is 19 years old.

In sports

In golf, the '19th hole' is the clubhouse bar and in match play, if there is a tie after 18 holes, an extra hole(s) is played. In miniature golf it is an extra hole on which the winner earns an instant prize.

References

↑ Sloane, N. J. A. (ed.). "Sequence A006512 (Greater of twin primes.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 5 ஆகத்து 2022.
↑ Sloane, N. J. A. (ed.). "Sequence A088762 (Numbers n such that (2n-1, 2n+3) is a cousin prime pair.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 5 ஆகத்து 2022.
↑ Sloane, N. J. A. (ed.). "Sequence A046117 (Primes p such that p-6 is also prime. (Upper of a pair of sexy primes.))". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 5 ஆகத்து 2022.
↑ Sloane, N. J. A. (ed.). "Sequence A002426 (Central trinomial coefficients: largest coefficient of (1 + x + x^2)^n.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 5 ஆகத்து 2022.
↑ Sloane, N. J. A. (ed.). "Sequence A002804 ((Presumed) solution to Waring's problem.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 5 ஆகத்து 2022.
↑ Sloane, N. J. A. (ed.). "Sequence A032020 (Number of compositions of n such that no two adjacent parts are equal (Carlitz compositions).)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 14 சூலை 2024.
↑ Sloane, N. J. A. (ed.). "Sequence A016038 (Strictly non-palindromic numbers: n is not palindromic in any base b with 2 less than or equal to b less than or equal to n-2.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 19 சூன் 2024.
↑ Sloane, N. J. A. (ed.). "Sequence A005900 (Octahedral numbers)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 17 ஆகத்து 2016.
↑ Sloane, N. J. A. (ed.). "Sequence A006784 (Engel expansion of Pi.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 14 சூன் 2024.
↑ "Sloane's A000043 : Mersenne exponents". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 மே 2016.
↑ Sloane, N. J. A. (ed.). "Sequence A007629 (Repfigit (REPetitive FIbonacci-like diGIT) numbers (or Keith numbers).)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 5 ஆகத்து 2022.
↑ Sloane, N. J. A. (ed.). "Sequence A001913 (Full reptend primes: primes with primitive root 10.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 5 ஆகத்து 2022.
↑ ^13.0 ^13.1 Sloane, N. J. A. (ed.). "Sequence A060544 (Centered 9-gonal (also known as nonagonal or enneagonal) numbers. Every third triangular number, starting with 1)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 30 நவம்பர் 2022.
↑ "19". Prime Curios!. Retrieved 5 ஆகத்து 2022.
↑ Sloane, N. J. A. (ed.). "Sequence A005349 (Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 11 அக்டோபர் 2022.
↑ Sloane, N. J. A. (ed.). "Sequence A051624 (12-gonal (or dodecagonal) numbers: a(n) equal to n*(5*n-4).)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 21 திசம்பர் 2023.
↑ Sloane, N. J. A. (ed.). "Sequence A068174 (Define an increasing sequence as follows. Start with an initial term, the seed (which need not have the property of the sequence); subsequent terms are obtained by inserting/placing at least one digit in the previous term to obtain the smallest number with the given property. Here the property is be a prime.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 26 சூலை 2022.
↑ Sloane, N. J. A. (ed.). "Sequence A088275 (Numbers n such that 10^n + 9 is prime)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 28 சூலை 2022.
↑ Guy, Richard; Unsolved Problems in Number Theory, p. 7 பன்னாட்டுத் தரப்புத்தக எண் 1475717385
↑ Sloane, N. J. A. (ed.). "Sequence A111441 (Numbers k such that the sum of the squares of the first k primes is divisible by k)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 2 சூன் 2022.
↑ "Sloane's A125602 : Centered triangular numbers that are prime". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 மே 2016.
↑ "Sloane's A003215 : Hex (or centered hexagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 மே 2016.
↑ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 13 சூலை 2022.
↑ Sloane, N. J. A. "Sequence A186076". The On-line Encyclopedia of Integer Sequences. Retrieved 13 சூலை 2022. Note that terms A186074(4) and A186074(10) have trailing 0's, i.e. 19900 = Sum_{k=0..199} k and 1999000 = Sum_{k=0..1999} k...". "This pattern continues indefinitely: 199990000, 19999900000, etc.
↑ Sloane, N. J. A. (ed.). "Sequence A055558 (Primes of the form 1999...999)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 26 சூலை 2022.
↑ Sloane, N. J. A. (ed.). "Sequence A007569 (Number of nodes in regular n-gon with all diagonals drawn.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 4 ஏப்ரல் 2023.
↑ Trigg, C. W. (பெப்ரவரி 1964). "A Unique Magic Hexagon". Recreational Mathematics Magazine. Retrieved 14 சூலை 2022.
↑ Martin Gardner (January 2012). "Hexaflexagons". The College Mathematics Journal (Taylor & Francis) 43 (1): 2–5. doi:10.4169/college.math.j.43.1.002.
↑ Sloane, N. J. A. (ed.). "Sequence A006534 (Number of one-sided triangular polyominoes (n-iamonds) with n cells; turning over not allowed, holes are allowed.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 8 திசம்பர் 2023.
↑ Andrews, William Symes (1917). Magic Squares and Cubes (PDF). Chicago, IL: Open Court Publishing Company. pp. 176, 177. ISBN 9780486206585. MR 0114763. கணினி நூலகம் 1136401. Zbl 1003.05500.
↑ Sloane, N. J. A. (ed.). "Sequence A072359 (Primes p such that the p-1 digits of the decimal expansion of k/p (for k equal to 1,2,3,...,p-1) fit into the k-th row of a magic square grid of order p-1.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 4 செப்டெம்பர் 2023.
↑ Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 6 செப்டெம்பர் 2023.
↑ Sloane, N. J. A. (ed.). "Sequence A006003 (a(n) equal to n*(n^2 + 1)/2.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 4 செப்டெம்பர் 2023.
↑ Sloane, N. J. A. "3x+1 problem". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 24 சனவரி 2023.
↑ Sloane, N. J. A. (ed.). "Sequence A006577 (Number of halving and tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 24 சனவரி 2023.
"Table of n, a(n) for n = 1..10000".
↑ Harold Scott MacDonald Coxeter (1982). "Ten toroids and fifty-seven hemidodecahedra". Geometriae Dedicata 13 (1): 87–99. doi:10.1007/BF00149428.
↑ Allcock, Daniel (11 July 2006). "Infinitely many hyperbolic Coxeter groups through dimension 19". Geometry & Topology 10 (2): 737–758. doi:10.2140/gt.2006.10.737.
↑ Tumarkin, P. (2004). "Hyperbolic Coxeter n-polytopes with n + 2 facets". Mathematical Notes (Springer) 75 (5/6): 848–854. doi:10.1023/B:MATN.0000030993.74338.dd.
↑ Seigal, Anna (2020). "Ranks and symmetric ranks of cubic surfaces". Journal of Symbolic Computation (Amsterdam: Elsevier) 101: 304–306. doi:10.1016/j.jsc.2019.10.001. Bibcode: 2018arXiv180105377S. https://www.sciencedirect.com/science/article/abs/pii/S074771711930104X.
↑ Sloane, N. J. A. (ed.). "Sequence A002267 (The 15 supersingular primes.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 11 திசம்பர் 2022.
↑ Ronan, Mark (2006). Symmetry and the Monster: One of the Greatest Quests of Mathematics. New York: Oxford University Press. pp. 244–246. doi:10.1007/s00283-008-9007-9. ISBN 978-0-19-280722-9. MR 2215662. கணினி நூலகம் 180766312. Zbl 1113.00002.
↑ Wilson, R.A (1998). "Chapter: An Atlas of Sporadic Group Representations" (PDF). The Atlas of Finite Groups - Ten Years On (LMS Lecture Note Series 249). Cambridge, U.K: Cambridge University Press. p. 267. doi:10.1017/CBO9780511565830.024. ISBN 9780511565830. கணினி நூலகம் 726827806. S2CID 59394831. Zbl 0914.20016.
List of standard generators of all sporadic groups.
↑ Nickerson, S.J.; Robert Anton Wilson (2011). "Semi-Presentations for the Sporadic Simple Groups". Experimental Mathematics (Oxfordshire: Taylor & Francis) 14 (3): 365. doi:10.1080/10586458.2005.10128927. https://www.tandfonline.com/doi/abs/10.1080/10586458.2005.10128927.
↑ Jansen, Christoph (2005). "The Minimal Degrees of Faithful Representations of the Sporadic Simple Groups and their Covering Groups". LMS Journal of Computation and Mathematics (London Mathematical Society) 8: 122−144. doi:10.1112/S1461157000000930.
↑ Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 28 பெப்ரவரி 2024.
↑ Sloane, N. J. A. (ed.). "Sequence A000292". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 28 பெப்ரவரி 2024.
↑ Sloane, N. J. A. (ed.). "Sequence A051871 (19-gonal (or enneadecagonal) numbers: n(17n-15)/2.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 9 திசம்பர் 2023.
↑ John F.R. Duncan; Michael H. Mertens; Ken Ono (2017). "Pariah moonshine". Nature Communications 8 (1): 2 (Article 670). doi:10.1038/s41467-017-00660-y. பப்மெட்:28935903. Bibcode: 2017NatCo...8..670D. "...so [sic] moonshine illuminates a physical origin for the monster, and for the 19 other sporadic groups that are involved in the monster.".
↑ R. B. Howlett; L. J. Rylands; D. E. Taylor (2001). "Matrix generators for exceptional groups of Lie type". Journal of Symbolic Computation 31 (4): 429. doi:10.1006/jsco.2000.0431. "...for all groups of Lie type, including the twisted groups of Steinberg, Suzuki and Ree (and the Tits group).".
↑ "Sloane's A003173 : Heegner numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 மே 2016.
↑ Brigid: Triple Goddess of the Flame (Health, Hearth, & Forge)

External links

விக்சனரியில் nineteen என்னும் சொல்லைப் பார்க்கவும்.

தலைப்பு=

Mukund Varadarajan
Major; Mukund Varadarajan; AC
பிறப்பு	வார்ப்புரு:Birth date text; Avadi, Chennai, Tamil Nadu, India
இறப்பு	25 ஏப்ரல் 2014 (அகவை 31); Shopian district,; Jammu & Kashmir State, India
அடக்கம்	Besant Nagar, Chennai
சார்பு	India
சேவை/கிளை	Indian Army
சேவைக்காலம்	2006 - 2014
தரம்	Major
படைப்பிரிவு	22 Rajput Regiment ; 44 Rashtriya Rifles (Rajput) battalion (deputation)
போர்கள்/யுத்தங்கள்	United Nations Lebanon Mission; Insurgency in Jammu and Kashmir †;
விருதுகள்	Ashoka Chakra
கல்வி	Sri Chandrasekharendra Saraswathi Viswa Mahavidyalaya (BCom); Madras Christian College (DJMC); Officers Training Academy;
துணை(கள்)	Indhu Rebecca Varghese
பிள்ளைகள்	1

	← 10 11 12 13 14 15 16 17 18 19 → எண்களின் பட்டியல் — முழு எண்கள் ← 0 10 20 30 40 50 60 70 80 90 →
முதலெண்	nineteen
வரிசை	19-ஆம்; (பத்தொன்பதாம்)
எண்ணுரு	nonadecimal
காரணியாக்கல்	prime
பகா எண்	8th
காரணிகள்	1, 19
ரோமன்	XIX
இரும எண்	100112
முன்ம எண்	2013
நான்ம எண்	1034
ஐம்ம எண்	345
அறும எண்	316
எண்ணெண்	238
பன்னிருமம்	1712
பதினறுமம்	1316
இருபதின்மம்	J20
36ம்ம எண்	J36
Hebrew numeral	י"ט
Babylonian numeral	𒌋𒐝

Major Mukund Varadarajan AC (12 April 1983 – 25 April 2014) was an Indian Army officer and a recipient of the Ashoka Chakra. Mukund, a commissioned officer in the Indian Army's Rajput Regiment, was posthumously awarded the Ashok Chakra for his actions during a counterterrorism operation while on deputation to the 44th Rashtriya Rifles battalion in Jammu and Kashmir. His biographical film in Tamil, Amaran, was released on the occasion of Diwali, 31 October 2024.

Early life and education

Mukund Varadarajan^[2] was born on 12 April 1983 to a Tamil speaking iyengar family^[1] to R Varadarajan and Geetha Varadarajan in his native Paruthipattu Village, Avadi, Chennai, Tamil Nadu, India.^[5] Mukund and his family moved to Tambaram after his father got job in a Public Sector Bank.^[5] His grandfather Raghavachari and two of his uncles also served in the army and this motivated him to join the Armed Forces.^[1]^[6]

Mukund received his Bachelor of Commerce From Sri Chandrasekharendra Saraswathi Viswa Mahavidyalaya at Kanchipuram and a diploma in journalism from the Madras Christian College, Chennai.^[1]^[6]

Mukund has two sisters, Swetha and Nithya. He married his longtime-girlfriend and college mate Indhu Rebecca Varghese on 28 August 2009. The couple's daughter, Arshea Mukund, was born on 17 March 2011.^[7]^[2]^[8]^[6]

Military career

Mukund was an alumnus of Officers Training Academy,^[2] and after graduating, he received a short service commission as a Lieutenant in the Rajput Regiment (22 Rajput) on 18 March 2006.^[9] On 18 March 2006, he was granted a regular commission with the rank of lieutenant,^[10] and received an ante-dated promotion to Captain from 18 October 2008.^[11] He served at the Infantry School in Mhow, Madhya Pradesh and was part of a United Nations Mission in Lebanon.^[1] He was promoted to Major on 18 October 2012 and was deputed to the 44th Battalion of the Rashtriya Rifles that December and posted in the Shopian district of Jammu and Kashmir.^[12]^[6]

Qazipathri operation

On 25 April 2014, Mukund led a cordon and search operation in a village in South Kashmir, based on intelligence inputs regarding the presence of militants. During the operation, his team came under heavy fire. Mukund quickly responded, aware that the cover of darkness could allow the militants to escape. Since the number of civilians inside the house was unclear, the use of heavy-caliber weapons was not an option. Along with Sepoy Vikram Singh, Mukund crawled through the orchard in front of the house, avoiding gunfire. After successfully crossing the orchard, they entered the house, where they were met with intense firing, and Mukund was grazed by a bullet on his forearm. The two soldiers swiftly retaliated, killing one of the militants. Mukund identified that the militant commander they were searching for was not among the dead. The remaining two militants, including a senior commander of Hizbul Mujahideen, attempted to escape using grenade explosions. Mukund and Singh took cover to avoid injury. In the chaos, Mukund recognized the senior commander among the fleeing militants as they made their way to the nearby outhouse.

Vikram Singh and his comrade approached the cement outhouse, believing that a grenade deployed moments earlier had neutralized the threat inside. However, as they stormed the structure, a sudden volley of gunfire hit Vikram. It became clear that while the grenade had indeed killed one individual, another remained. The commander, who had fled from the debris of a nearby residence, had not been alone. The second individual had been killed by the grenade, but the commander, Asif Wani, survived.

As the rounds of fire were exchanged, Wani, shielded by a row of logs inside the outhouse, continued to fire from his confined position. Vikram Singh, having taken two bullets—one to the neck and another to the jaw—collapsed. His comrade, Mukund, understood the severity of the wounds and realized that Vikram's life was at risk. Despite the gravity of the situation, there was no time for hesitation. Mukund immediately advanced, firing his AK-47 at Asif Wani, fatally wounding him.

Though some of Asif Wani's shots hit Mukund in the process, he initially appeared unharmed as he emerged from the outhouse. An officer in the surrounding cordon noted, "He looked okay. We thought he was fine." But shortly after, Mukund collapsed, having sustained three gunshot wounds. He lost consciousness and, despite efforts to evacuate him, succumbed to his injuries.^[13]^[14]^[15]

For his actions during the operation, he was posthumously awarded the Ashok Chakra, India's highest peacetime gallantry award in 2014, for displaying valor beyond the call of duty.^[13] The citation by the Government of India during the declaration of the Ashok Chakra read:^[2]

During the operation, before attaining martyrdom, Major Mukund displayed exemplary leadership skills, raw courage, planning and swift action, which culminated in the elimination of the three top ranked Hizbul Mujaheddin terrorists.

Mukund is the fourth recipient of the Ashoka Chakra from the state of Tamil Nadu.

On June 1, 2015, a bust of Major Mukund Varadarajan was unveiled in honor of his patriotism and sacrifice. It is displayed at the Officers Training Academy.^[16]

பதவிகளும் தேதியும்

பதவி	பணி	பதவியின் தேதி
லெப்டினண்ட்	இந்திய இராணுவம்	18 மார்ச் 2006
கேப்டன்	இந்திய இராணுவம்	18 அக்டோபர் 2008
மேஜர்	இந்திய இராணுவம்	18 அக்டோபர் 2012

Legacy

A bust of Major Mukund was unveiled by his Wife Indu Verghese at Rajput Regiment.^[17] Mukund's family also unveiled a statue of him at Officers Training Academy, Chennai.^[18]

The 2015 Malayalam-language anti-war film Picket 43 was dedicated as a tribute to Major Varadarajan,^[19] who had been assisting the film crew during production. However, he was killed in combat with militants in the Shopian area while filming was underway in Jammu and Kashmir.^[20] In 2024, a Tamil biopic, Amaran, was announced. It is based on Mukund's life and the 44 Rashtriya Rifles' fight against terrorism. Sivakarthikeyan portrayed Major Mukund Varadarajan and his wife Indu Rebecca Varghese was played by Sai Pallavi.^[21]^[22]^[23]^[24]^[25]

References

↑ ^1.0 ^1.1 ^1.2 ^1.3 ^1.4 "Tambaram mourns a braveheart". The Hindu. 27 April 2014. http://www.thehindu.com/news/cities/chennai/tambaram-mourns-a-braveheart/article5951633.ece.
↑ ^2.0 ^2.1 ^2.2 ^2.3 ^2.4 "மேஜர் முகுந்த் வரதராஜன் யார்? அவரது கடைசி தருணம் எப்படி இருந்தது?". https://www.bbc.com/tamil/articles/cdd092qlglpo.
↑ "Major Mukund spouse:Indhu Rebecca Varadharajan's final journey". The Hindu. 28 April 2014. http://www.thehindu.com/news/cities/chennai/major-mukund-varadharajans-final-journey/article5957128.ece#im-image-6.
↑ ^4.0 ^4.1 "'Love my son… I am proud of him… I miss him'". 28 ஏப்ரல் 2014.
↑ ^5.0 ^5.1 "அமரன்: மேஜர் முகுந்த் வரதராஜன் யார்? அவரது கடைசி தருணம் எப்படி இருந்தது?". BBC News தமிழ். 30 அக்டோபர் 2024. Retrieved 3 நவம்பர் 2024.
↑ ^6.0 ^6.1 ^6.2 ^6.3 "Major Worked at BPO Before Realising Childhood Dream". The New Indian Express. 27 April 2014 இம் மூலத்தில் இருந்து 28 April 2014 அன்று. பரணிடப்பட்டது.. https://web.archive.org/web/20140428215359/http://www.newindianexpress.com/cities/chennai/Major-Worked-at-BPO-Before-Realising-Childhood-Dream/2014/04/27/article2191625.ece.
↑ "'India Should See The Man Mukund Was, Not My Sorrow': Martyr's Wife to NDTV". NDTV. Retrieved 31 சனவரி 2015.
↑ "Major Mukund Varadarajan cremated with full military honours in Chennai". The Times of India. 29 April 2014. http://timesofindia.indiatimes.com/india/Major-Mukund-Varadarajan-cremated-with-full-military-honours-in-Chennai/articleshow/34347210.cms.
↑ "Part I-Section 4: Ministry of Defence (Army Branch)". The Gazette of India. 21 April 2007. p. 625. http://egazette.nic.in/WriteReadData/2007/W_16_2011_060.pdf.
↑ "Part I-Section 4: Ministry of Defence (Army Branch)". The Gazette of India. 11 June 2011. p. 1101. http://egazette.nic.in/WriteReadData/2011/W_24_2011_088.pdf.
↑ "Part I-Section 4: Ministry of Defence (Army Branch)". The Gazette of India. 3 September 2011. p. 1622. http://egazette.nic.in/WriteReadData/2011/W_36_2011_098.pdf.
↑ "Part I-Section 4: Ministry of Defence (Army Branch)". The Gazette of India. 13 July 2013. p. 1058. http://egazette.nic.in/WriteReadData/2013/W_28_2013_322.pdf.
↑ ^13.0 ^13.1 "For Gallantry". 14 ஆகத்து 2014. Retrieved 16 ஆகத்து 2014.
↑ "Major Mukund Varadarajan Awarded Ashok Chakra, the Highest Gallantry Award". NDTV. 14 August 2014. http://www.ndtv.com/article/india/major-mukund-varadarajan-awarded-ashok-chakra-the-highest-gallantry-award-576105.
↑ "Independence Day Gallantry Awards and Other Decorations". PIB. 14 August 2014. http://pib.nic.in/newsite/PrintRelease.aspx?relid=108722.
↑ Desk, OneIndia Defence (1 சூன் 2015). "Bust in memory of Major Mukund; Capt Kalia a forgotten hero". Oneindia (in ஆங்கிலம்). Retrieved 17 பெப்ரவரி 2020. {{cite web}}: |last= has generic name (help)
↑ "जम्मू के शोपियन जिले में आतंकियों से मोर्चा लेते हुए शहीद हुए आरआरसी के मेजर मुकुंद वरदराजन की प्रतिमा का अनावरण शनिवार को राजपूत रेजीमेंट में उनकी पत्नी इंदू वरदराजन ने किया।". Amar Ujala. https://www.amarujala.com/amp/uttar-pradesh/farrukhabad/the-unveiled-the-of-the-statue-of-martyr-major-mukund-varadarajan-hindi-news.
↑ "Major Mukund Varadarajan's bust unveiled at Officers Training Academy". Mathrubhumi. https://www.google.com/english.mathrubhumi.com/amp/news/india/major-mukund-varadarajan-s-bust-unveiled-at-officers-training-academy-1.263833.
↑ Ramachandran, Mythily (11 February 2015). "'Picket 43,' a tale of friendship beyond borders". Gulf News. https://gulfnews.com/entertainment/south-indian/picket-43-a-tale-of-friendship-beyond-borders-1.1454497.
↑ Karthikeyan, Shruti (1 July 2014). "I have fought and killed people, but fighting is not a solution: Major Ravi". The Times of India. https://timesofindia.indiatimes.com/entertainment/malayalam/movies/news/i-have-fought-and-killed-people-but-fighting-is-not-a-solution-major-ravi/articleshow/37526599.cms.
↑ Bureau, The Hindu (2024-02-16). "'Amaran' teaser: Sivakarthikeyan as Major Mukund Varadarajan prepares his men for battle" (in en-IN). The Hindu. https://www.thehindu.com/entertainment/movies/amaran-teaser-sivakarthikeyan-as-major-mukund-varadarajan-prepares-his-men-for-battle/article67853675.ece.
↑ "Amaran: Sivakarthikeyan's title teaser & all about Major Mukund Varadarajan". www.moviecrow.com. Retrieved 17 பெப்ரவரி 2024.
↑ "Kamal Haasan unveils Sivakarthikeyan's 'Amaran' teaser ahead of his birthday". India Today (in ஆங்கிலம்). 16 பெப்ரவரி 2024. Retrieved 17 பெப்ரவரி 2024.
↑ "Sivakarthikeyan Plays Real Life Hero Major Mukund Varadarajan In His Next Film Amaran". TimesNow (in ஆங்கிலம்). 16 பெப்ரவரி 2024. Retrieved 17 பெப்ரவரி 2024.
↑ "SK21 titled Amaran: Sivakarthikeyan says 'honored to have portrayed Major Mukund Varadarajan' as he unveils first teaser". PINKVILLA (in ஆங்கிலம்). 16 பெப்ரவரி 2024. Retrieved 17 பெப்ரவரி 2024.

வார்ப்புரு:Ashoka Chakra award recipients

எச்சரிக்கை: இயல்புநிலை வரிசைப்படுத்து விசை Varadarajan, Mukund " முன்னால் இயல்புநிலை வரிசைப்படுத்து விசை "19 (Number)" ஐ மீறுகிறது.

[1] Sloane, N. J. A. (ed.). "Sequence A006512 (Greater of twin primes.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 5 ஆகத்து 2022.

[2] Sloane, N. J. A. (ed.). "Sequence A088762 (Numbers n such that (2n-1, 2n+3) is a cousin prime pair.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 5 ஆகத்து 2022.

[3] Sloane, N. J. A. (ed.). "Sequence A046117 (Primes p such that p-6 is also prime. (Upper of a pair of sexy primes.))". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 5 ஆகத்து 2022.

[4] Sloane, N. J. A. (ed.). "Sequence A002426 (Central trinomial coefficients: largest coefficient of (1 + x + x^2)^n.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 5 ஆகத்து 2022.

[5] Sloane, N. J. A. (ed.). "Sequence A002804 ((Presumed) solution to Waring's problem.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 5 ஆகத்து 2022.

[6] Sloane, N. J. A. (ed.). "Sequence A032020 (Number of compositions of n such that no two adjacent parts are equal (Carlitz compositions).)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 14 சூலை 2024.

[7] Sloane, N. J. A. (ed.). "Sequence A016038 (Strictly non-palindromic numbers: n is not palindromic in any base b with 2 less than or equal to b less than or equal to n-2.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 19 சூன் 2024.

[8] Sloane, N. J. A. (ed.). "Sequence A005900 (Octahedral numbers)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 17 ஆகத்து 2016.

[9] Sloane, N. J. A. (ed.). "Sequence A006784 (Engel expansion of Pi.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 14 சூன் 2024.

[:0-10] "Sloane's A000043 : Mersenne exponents". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 மே 2016.

[11] Sloane, N. J. A. (ed.). "Sequence A007629 (Repfigit (REPetitive FIbonacci-like diGIT) numbers (or Keith numbers).)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 5 ஆகத்து 2022.

[12] Sloane, N. J. A. (ed.). "Sequence A001913 (Full reptend primes: primes with primitive root 10.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 5 ஆகத்து 2022.

[Centered_nonagonal-13] 13.0 ^13.1 Sloane, N. J. A. (ed.). "Sequence A060544 (Centered 9-gonal (also known as nonagonal or enneagonal) numbers. Every third triangular number, starting with 1)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 30 நவம்பர் 2022.

[14] "19". Prime Curios!. Retrieved 5 ஆகத்து 2022.

[15] Sloane, N. J. A. (ed.). "Sequence A005349 (Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 11 அக்டோபர் 2022.

[16] Sloane, N. J. A. (ed.). "Sequence A051624 (12-gonal (or dodecagonal) numbers: a(n) equal to n*(5*n-4).)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 21 திசம்பர் 2023.

[17] Sloane, N. J. A. (ed.). "Sequence A068174 (Define an increasing sequence as follows. Start with an initial term, the seed (which need not have the property of the sequence); subsequent terms are obtained by inserting/placing at least one digit in the previous term to obtain the smallest number with the given property. Here the property is be a prime.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 26 சூலை 2022.

[18] Sloane, N. J. A. (ed.). "Sequence A088275 (Numbers n such that 10^n + 9 is prime)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 28 சூலை 2022.

[19] Guy, Richard; Unsolved Problems in Number Theory, p. 7 பன்னாட்டுத் தரப்புத்தக எண் 1475717385

[20] Sloane, N. J. A. (ed.). "Sequence A111441 (Numbers k such that the sum of the squares of the first k primes is divisible by k)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 2 சூன் 2022.

[21] "Sloane's A125602 : Centered triangular numbers that are prime". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 மே 2016.

[22] "Sloane's A003215 : Hex (or centered hexagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 மே 2016.

[23] Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 13 சூலை 2022.

[24] Sloane, N. J. A. "Sequence A186076". The On-line Encyclopedia of Integer Sequences. Retrieved 13 சூலை 2022. Note that terms A186074(4) and A186074(10) have trailing 0's, i.e. 19900 = Sum_{k=0..199} k and 1999000 = Sum_{k=0..1999} k...". "This pattern continues indefinitely: 199990000, 19999900000, etc.

[25] Sloane, N. J. A. (ed.). "Sequence A055558 (Primes of the form 1999...999)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 26 சூலை 2022.

[26] Sloane, N. J. A. (ed.). "Sequence A007569 (Number of nodes in regular n-gon with all diagonals drawn.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 4 ஏப்ரல் 2023.

[27] Trigg, C. W. (பெப்ரவரி 1964). "A Unique Magic Hexagon". Recreational Mathematics Magazine. Retrieved 14 சூலை 2022.

[28] Martin Gardner (January 2012). "Hexaflexagons". The College Mathematics Journal (Taylor & Francis) 43 (1): 2–5. doi:10.4169/college.math.j.43.1.002.

[29] Sloane, N. J. A. (ed.). "Sequence A006534 (Number of one-sided triangular polyominoes (n-iamonds) with n cells; turning over not allowed, holes are allowed.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 8 திசம்பர் 2023.

[30] Andrews, William Symes (1917). Magic Squares and Cubes (PDF). Chicago, IL: Open Court Publishing Company. pp. 176, 177. ISBN 9780486206585. MR 0114763. கணினி நூலகம் 1136401. Zbl 1003.05500.

[31] Sloane, N. J. A. (ed.). "Sequence A072359 (Primes p such that the p-1 digits of the decimal expansion of k/p (for k equal to 1,2,3,...,p-1) fit into the k-th row of a magic square grid of order p-1.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 4 செப்டெம்பர் 2023.

[32] Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 6 செப்டெம்பர் 2023.

[33] Sloane, N. J. A. (ed.). "Sequence A006003 (a(n) equal to n*(n^2 + 1)/2.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 4 செப்டெம்பர் 2023.

[34] Sloane, N. J. A. "3x+1 problem". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 24 சனவரி 2023.

[35] Sloane, N. J. A. (ed.). "Sequence A006577 (Number of halving and tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 24 சனவரி 2023.
"Table of n, a(n) for n = 1..10000".

[36] Harold Scott MacDonald Coxeter (1982). "Ten toroids and fifty-seven hemidodecahedra". Geometriae Dedicata 13 (1): 87–99. doi:10.1007/BF00149428.

[37] Allcock, Daniel (11 July 2006). "Infinitely many hyperbolic Coxeter groups through dimension 19". Geometry & Topology 10 (2): 737–758. doi:10.2140/gt.2006.10.737.

[38] Tumarkin, P. (2004). "Hyperbolic Coxeter n-polytopes with n + 2 facets". Mathematical Notes (Springer) 75 (5/6): 848–854. doi:10.1023/B:MATN.0000030993.74338.dd.

[39] Seigal, Anna (2020). "Ranks and symmetric ranks of cubic surfaces". Journal of Symbolic Computation (Amsterdam: Elsevier) 101: 304–306. doi:10.1016/j.jsc.2019.10.001. Bibcode: 2018arXiv180105377S. https://www.sciencedirect.com/science/article/abs/pii/S074771711930104X.

[40] Sloane, N. J. A. (ed.). "Sequence A002267 (The 15 supersingular primes.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 11 திசம்பர் 2022.

[41] Ronan, Mark (2006). Symmetry and the Monster: One of the Greatest Quests of Mathematics. New York: Oxford University Press. pp. 244–246. doi:10.1007/s00283-008-9007-9. ISBN 978-0-19-280722-9. MR 2215662. கணினி நூலகம் 180766312. Zbl 1113.00002.

[42] Wilson, R.A (1998). "Chapter: An Atlas of Sporadic Group Representations" (PDF). The Atlas of Finite Groups - Ten Years On (LMS Lecture Note Series 249). Cambridge, U.K: Cambridge University Press. p. 267. doi:10.1017/CBO9780511565830.024. ISBN 9780511565830. கணினி நூலகம் 726827806. S2CID 59394831. Zbl 0914.20016.
List of standard generators of all sporadic groups.

[43] Nickerson, S.J.; Robert Anton Wilson (2011). "Semi-Presentations for the Sporadic Simple Groups". Experimental Mathematics (Oxfordshire: Taylor & Francis) 14 (3): 365. doi:10.1080/10586458.2005.10128927. https://www.tandfonline.com/doi/abs/10.1080/10586458.2005.10128927.

[44] Jansen, Christoph (2005). "The Minimal Degrees of Faithful Representations of the Sporadic Simple Groups and their Covering Groups". LMS Journal of Computation and Mathematics (London Mathematical Society) 8: 122−144. doi:10.1112/S1461157000000930.

[45] Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 28 பெப்ரவரி 2024.

[46] Sloane, N. J. A. (ed.). "Sequence A000292". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 28 பெப்ரவரி 2024.

[47] Sloane, N. J. A. (ed.). "Sequence A051871 (19-gonal (or enneadecagonal) numbers: n(17n-15)/2.)". நேரிணைய எண்வரிசை கலைக்களஞ்சியம். நேரிணைய எண்வரிசை கலைக்களஞ்சிய அறக்கட்டளை. Retrieved 9 திசம்பர் 2023.

[48] John F.R. Duncan; Michael H. Mertens; Ken Ono (2017). "Pariah moonshine". Nature Communications 8 (1): 2 (Article 670). doi:10.1038/s41467-017-00660-y. பப்மெட்:28935903. Bibcode: 2017NatCo...8..670D. "...so [sic] moonshine illuminates a physical origin for the monster, and for the 19 other sporadic groups that are involved in the monster.".

[49] R. B. Howlett; L. J. Rylands; D. E. Taylor (2001). "Matrix generators for exceptional groups of Lie type". Journal of Symbolic Computation 31 (4): 429. doi:10.1006/jsco.2000.0431. "...for all groups of Lie type, including the twisted groups of Steinberg, Suzuki and Ree (and the Tits group).".

[50] "Sloane's A003173 : Heegner numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 மே 2016.

[51] Brigid: Triple Goddess of the Flame (Health, Hearth, & Forge)

[thin2-52] 1.0 ^1.1 ^1.2 ^1.3 ^1.4 "Tambaram mourns a braveheart". The Hindu. 27 April 2014. http://www.thehindu.com/news/cities/chennai/tambaram-mourns-a-braveheart/article5951633.ece.

[dc-53] 2.0 ^2.1 ^2.2 ^2.3 ^2.4 "மேஜர் முகுந்த் வரதராஜன் யார்? அவரது கடைசி தருணம் எப்படி இருந்தது?". https://www.bbc.com/tamil/articles/cdd092qlglpo.

[thehindu1-54] "Major Mukund spouse:Indhu Rebecca Varadharajan's final journey". The Hindu. 28 April 2014. http://www.thehindu.com/news/cities/chennai/major-mukund-varadharajans-final-journey/article5957128.ece#im-image-6.

[indianexpress.com-55] 4.0 ^4.1 "'Love my son… I am proud of him… I miss him'". 28 ஏப்ரல் 2014.

[:0-56] 5.0 ^5.1 "அமரன்: மேஜர் முகுந்த் வரதராஜன் யார்? அவரது கடைசி தருணம் எப்படி இருந்தது?". BBC News தமிழ். 30 அக்டோபர் 2024. Retrieved 3 நவம்பர் 2024.

[tniex1-57] 6.0 ^6.1 ^6.2 ^6.3 "Major Worked at BPO Before Realising Childhood Dream". The New Indian Express. 27 April 2014 இம் மூலத்தில் இருந்து 28 April 2014 அன்று. பரணிடப்பட்டது.. https://web.archive.org/web/20140428215359/http://www.newindianexpress.com/cities/chennai/Major-Worked-at-BPO-Before-Realising-Childhood-Dream/2014/04/27/article2191625.ece.

[58] "'India Should See The Man Mukund Was, Not My Sorrow': Martyr's Wife to NDTV". NDTV. Retrieved 31 சனவரி 2015.

[59] "Major Mukund Varadarajan cremated with full military honours in Chennai". The Times of India. 29 April 2014. http://timesofindia.indiatimes.com/india/Major-Mukund-Varadarajan-cremated-with-full-military-honours-in-Chennai/articleshow/34347210.cms.

[60] "Part I-Section 4: Ministry of Defence (Army Branch)". The Gazette of India. 21 April 2007. p. 625. http://egazette.nic.in/WriteReadData/2007/W_16_2011_060.pdf.

[61] "Part I-Section 4: Ministry of Defence (Army Branch)". The Gazette of India. 11 June 2011. p. 1101. http://egazette.nic.in/WriteReadData/2011/W_24_2011_088.pdf.

[62] "Part I-Section 4: Ministry of Defence (Army Branch)". The Gazette of India. 3 September 2011. p. 1622. http://egazette.nic.in/WriteReadData/2011/W_36_2011_098.pdf.

[63] "Part I-Section 4: Ministry of Defence (Army Branch)". The Gazette of India. 13 July 2013. p. 1058. http://egazette.nic.in/WriteReadData/2013/W_28_2013_322.pdf.

[stratpost-64] 13.0 ^13.1 "For Gallantry". 14 ஆகத்து 2014. Retrieved 16 ஆகத்து 2014.

[65] "Major Mukund Varadarajan Awarded Ashok Chakra, the Highest Gallantry Award". NDTV. 14 August 2014. http://www.ndtv.com/article/india/major-mukund-varadarajan-awarded-ashok-chakra-the-highest-gallantry-award-576105.

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