Английская Википедия:744 (number)

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744 (seven hundred [and] forty four) is the natural number following 743 and preceding 745.

744 plays a major role within moonshine theory of sporadic groups, in context of the classification of finite simple groups.

Number theory

744 is the nineteenth number of the form <math>pqr^{3}</math> where <math>r</math>, <math>p</math> and <math>q</math> represent distinct prime numbers (2, 3, and 31; respectively).

It can be represented as the sum of nonconsecutive factorials <math>k!</math>, as the sum of four consecutive primes <math>p</math>, and as the product of sums of divisors <math>\sigma(n)</math> of consecutive integers <math>n</math>; respectively:Шаблон:Efn-la

<math display=block> \begin{align} 744 & = 4! + 6! \\ 744 & = 179 + 181 + 191 + 193 \\ 744 & = \sigma(15) \times \sigma(16) = 24 \times 31 \\ \end{align}</math>

744 contains sixteen total divisors — fourteen aside from its largest and smallest unitary divisors — all of which collectively generate an integer arithmetic mean of <math>120 = 5!</math> that is also the first number of the form <math>pqr^{3}.</math>Шаблон:Efn-la

The number partitions of the square of seven (49) into prime parts is 744, as is the number of partitions of 48 into at most four distinct parts.Шаблон:Efn-la

It is palindromic in septenary (2112₇), while in binary it is a pernicious number, as its digit representation (1011101000₂) contains a prime count (5) of ones.Шаблон:Efn-la

It is also abundant and semiperfect, as well as practical. It is the first number to be the sum of nine cubes in eight or more ways,Шаблон:Efn-la and the number of six-digit perfect powers in decimal.

744 is theta series coefficient 25 of four-dimensional cubic lattice <math>\mathbb {D}_{4} \cong \mathbb {(Z^{4})^{+}} </math> associated with the ring of Hurwitz quaternions.Шаблон:Efn

Totients

744 has two hundred and forty integers that are relatively prime or coprime with and up to itself, equivalently its Euler totient.Шаблон:Efn-la

This totient of 744 is regular like its sum-of-divisors, where 744 sets the twenty-ninth record for <math>\sigma(n)</math> of 1920.Шаблон:Efn-la Both the totient and sum-of-divisors values of 744 contain the same set of distinct prime factors (2, 3, and 5), while the Carmichael function or reduced totient (which counts the least common multiple of order of elements in a multiplicative group of integers modulo <math>n</math>) at seven hundred forty-four is equal to <math>\lambda(744) = 30 = 2 \times 3 \times 5</math>.Шаблон:Efn-la 744 is also a Zumkeller number whose divisors can be partitioned into two disjoint sets with equal sum (960).Шаблон:Efn-la

Of these 240 totatives, 110 are strictly composite totatives that nearly match the sequence of composite numbers up to 744 that are congruent to <math>\pm {1} \bmod {6}</math>, which is the same congruence that all prime numbers greater than 3 hold.Шаблон:Efn-la Only seven numbers present in this sequence are not totatives of 744 (less-than); they are 713, 589, 527, 403, 341, 217, and 155; all of which are divisible by the eleventh prime number 31.Шаблон:Efn-la The remaining 130 totatives are 1 and all the primes between 5 and 743 except for 31 (all prime numbers less than 744 that are not part of its prime factorization) where its largest prime totative of 743 has a prime index of 132 (the smallest digit-reassembly number in decimal).Шаблон:Efn-la On the other hand, only three numbers hold a totient of 744; they are 1119, 1492, and 2238.Шаблон:Efn-la

744 is the sixth number <math>n</math> whose totient value has a sum of divisors equal to <math>n</math>: <math>\sigma(\varphi(744))=744</math>. A total of seven numbers have sums of divisors equal to 744, they are 240, 350, 366, 368, 575, 671, and 743.Шаблон:Efn-la Otherwise, the aliquot sum of 744, which represents the sum of all divisors of 744 aside from itself, is 1176 which is the forty-eighth triangular number, and the binomial coefficient <math>\operatorname {C(49,2)}</math> present inside the forty-ninth row of Pascal's triangle.Шаблон:Efn-la</math>, and the forty-first 6-almost prime that is divisible by exactly six primes with multiplicity. }} If only the fourteen proper divisors of 744 are considered, then the sum generated by these is 1175, whose six divisors contain an arithmetic mean of 248,Шаблон:Efn-la the fourteenth-largest divisor of 744. Only one number has an aliquot sum that is 744, it is 456.Шаблон:Efn-la

Graph theory

The number of Euler tours (or Eulerian cycles) of the complete, undirected graph <math>K_{6}</math> on six vertices and fifteen edges is 744. On seven vertices, there are 129,976,320 Euler tours. These can only be generated on complete graphs with at least three vertices; the number of tours for three, four and five vertices are respectively 2, 2 and 264 (the latter is the second digit-reassembly number in base ten).Шаблон:Efn-la On the other hand, the number of Euler tours of the complete digraph, or directed graph, on four vertices is 256, while on five vertices it is 972,000 (and 247,669,456,896 on six vertices), by the BEST theorem.

Otherwise, 745 is the number of disconnected simple labeled graphs covering six vertices, where the most symmetric of these graphs has three pairs of distinct vertices each covered by one edge alone and all three edges intersecting; this yields the disconnected covering graph <math>\{\{1,4\},\{2,5\},\{3,6\}\}</math> on vertices labelled <math>1</math> through <math>6</math> in a hexagonal arrangement, with the remaining 744 graphs representing all other possible conformations.

Convolution of Fibonacci numbers

744 is the twelfth self-convolution of Fibonacci numbers, which is equivalently the number of elements in all subsets of <math>\{1,2,3,...,11\}</math> with no consecutive integers.

Abstract algebra

The j–invariant holds as a Fourier series [[Modular form#Modular functions|Шаблон:Math–expansion]],Шаблон:Efn-lg <math display=block>j(\tau) = q^{-1} + 744 + 196\,884 q + 21\,493\,760 q^2 + 864\,299\,970 q^3 + \cdots</math>

The friendly giant <math>\mathrm {F_{1}}</math> contains an infinitely graded faithful dimensional representation equivalent to the <math>q</math> coefficients of this series, where <math>q = e^{2\pi i\tau}</math> and <math>\tau</math> the half-period ratio of an elliptic function.

Also, the almost integer

<math display=block>e^{\pi \sqrt{163}} \approx 262\,537\,412\,640\,768\,743.999\,999\,999\,999\,250\,072\,59 \approx 640\,320^3 + 744.</math>

This number is known as Ramanujan's constant, which is transcendental. Mark Ronan and other prominent mathematicians have noted that the appearance of <math>163</math> in this number is relevant within moonshine theory, where one hundred and sixty-three is the largest of nine Heegner numbers that are square-free positive integers <math>d</math> such that the imaginary quadratic field <math display="inline">\Q\left[\sqrt{-d}\right]</math> has class number of <math>1</math> (equivalently, the ring of integers over the same algebraic number field have unique factorization).Шаблон:Rp <math>\mathrm {F_{1}}</math> has one hundred and ninety-four (194) conjugacy classes generated from its character table that collectively produces the same number of elliptic moonshine functions which are not all linearly independent; only one hundred and sixty-three are entirely independent of one another. The linear term of error <math>O</math> for Ramajunan's constant is approximately,

<math display=block>\frac{-196\,884}{e^{\pi \sqrt{163}}} \approx \frac{-196\,884}{640\,320^3+744} \approx -0.000\,000\,000\,000\,75 ,</math>

where <math>196\,884</math> is the value of the minimal faithful complex dimensional representation of <math>\mathrm {F_{1}}</math>, the largest sporadic group.

Specifically, all three common prime factors <math>(2,3,5)</math> that divide the Euler totient, sum-of-divisors, and reduced totient of <math>744</math> are the smallest and only primes that divide the orders of all twenty-six sporadic groups, in contrast with only six groups <math>(\mathrm {F_{1}},\text{ }\mathrm {B},\text{ }\mathrm {F_{3}},\text{ }\mathrm {Ly},\text{ }\mathrm {ON},\text{ }\mathrm {J_{4}})</math> whose orders can be divided by the largest consecutive supersingular prime and largest prime factor of seven hundred and forty-four, <math>31</math>;Шаблон:RpШаблон:Efn-lg three of these belong inside the small family of six pariah groups that are not subquotients of <math>\mathrm {F_{1}}.</math> The largest supersingular prime that divides the order of <math>\mathrm {F_{1}}</math> is <math>71</math>, which is the eighth self-convolution of Fibonacci numbers, where <math>744</math> is the twelfth.Шаблон:Efn-lg

The largest three Heegner numbers with <math>d > 19</math> also give rise to almost integers of the form <math>e^{\pi \sqrt{d}}</math> which involve <math>744</math>. In increasing orders of approximation,Шаблон:RpШаблон:Efn-lg

<math display=block>\begin{align} e^{\pi \sqrt{19}} &\approx {\color{white}000\,0}96^3+744-0.22\\ e^{\pi \sqrt{43}} &\approx {\color{white}000\,}960^3+744-0.000\,22\\ e^{\pi \sqrt{67}} &\approx {\color{white}00}5\,280^3+744-0.000\,0013\\ e^{\pi \sqrt{163}} &\approx 640\,320^3+744-0.000\,000\,000\,000\,75 \end{align} </math>

Square-free positive integers over the negated imaginary quadratic field with class number of <math>2</math> also produce almost integers for values of <math>d</math>, where for instance there is <math>199\,148\,648^2 - 0.000\,97\ldots \approx e^{\pi \sqrt{148}} + (8 \times 10^{3} + 744).</math>Шаблон:Efn-lg

E₈ and the Leech lattice

Within finite simple groups of Lie type, exceptional Lie algebra <math>\mathfrak{e_{8}}</math> holds a minimal faithful representation in two hundred and forty-eight dimensions, where <math>248</math> divides <math>744</math> thrice over.Шаблон:Rp John McKay noted an intersection between finite simple groups of Lie type and those that are sporadic, where symmetries of nodes in the Dynkin diagrams of complex Lie algebra <math>\mathfrak{e}_{8}</math> as well as those of <math>\mathfrak{e}_{7}</math> and <math>\mathfrak{e}_{6}</math> respectively coincide with the three largest conjugacy classes of <math>\mathrm {F_{1}}</math>; where also the corresponding McKay–Thompson series <math>j(3\tau)^{1/3}</math> of sporadic Thompson group <math>\mathrm {Th}</math> holds coefficients representative of its faithful dimensional representation — also minimal at <math>\operatorname {dim} 248</math> — whose values themselves embed irreducible representation of <math>\mathfrak{e_{8}}</math>.Шаблон:Rp In turn, exceptional Lie algebra <math>\mathfrak{e_{8}}</math> is shown to have a graded dimension <math>j(q)^{1/3}</math> whose character <math>\chi</math> lends to a direct sum equivalent to,Шаблон:Rp

<math>\chi_{e_{8} \oplus e_{8} \oplus e_{8}} (q) = J(q) + 744 = j(q),</math> where the CFT probabilistic partition function for <math>\mathrm {F_{1}}</math> is <math>J(q)</math> of character <math>\chi_{F_{1}}.</math>

The twenty-four dimensional Leech lattice <math>\Lambda_{24}</math> in turn can be constructed using three copies of the associated <math>\mathbb {E_{8}}</math> latticeШаблон:RpШаблон:Efn-lg and with the eight-dimensional octonions <math>\mathbb {O}</math> (see also, Freudenthal magic square), where the automorphism group of <math>\mathbb {O}</math> is the smallest exceptional Lie algebra <math>\mathfrak{g_{2}}</math>, which embeds inside <math>\mathfrak{e_{8}}</math>. In the form of a vertex operator algebra, the Leech lattice VOA is the first aside from <math>V_1</math> (as <math>\mathbb {C}_{24}</math>) with a central charge <math>c</math> of <math>24</math>, out of a total seventy-one such modular invariant conformal field theories of holomorphic VOAs of weight one. Known as Schellekens' list, these algebras form deep holes in <math>V_{\Lambda_{24}}</math> whose corresponding orbifold constructions are isomorphic to the moonshine module <math>V_{2}</math>^{Шаблон:Large} that contains <math>\mathrm {F_{1}}</math> as its automorphism; of these, the second and third largest contain affine structures <math>E^3_{8,1}</math> and <math>D_{16,1}E_{8,1}</math> that are realized in <math>\operatorname {dim}744</math>.Шаблон:Efn-lg

Other properties

744 is also the sum of consecutive pentagonal numbers <math>P_{n}</math>,Шаблон:Efn-la <math display=block>744 = P_{11} + P_{12} + P_{13} = 210 + 247 + 287. </math>

It is also the magic constant of a six by six magic square consisting of thirty-six consecutive prime numbers, between 41 and 223 inclusive.Шаблон:Efn-la

744 is the number of non-congruent polygonal regions in a regular 36-gon with all diagonals drawn.

There are 744 ways in-which fourteen squares of different sizes fit edge-to-edge inside a larger rectangle.

See also

• Moonshine theory – the study between the relationships held by F₁ and modular functions, in particular j(τ)

• Heegner numbers — 1, 2, 3, 7, 11, 19, 43, 67, and 163
• Supersingular primes — 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31, as well as 41, 47, 59, and 71
• The j-invariant — 1728 = 12³ = j(i) , one less than the first taxicab number 1729 (also known as the Ramanujan–Hardy number)

• Osiris or digit-reassembly numbers – equal to the sum of all permutations of subsets of their own digits (e.g. 132 in decimal vis-à-vis symmetric group S₆)

Notes

Higher arithmetic

Heegner numbers, E₈ and the Leech lattice

References

Шаблон:Reflist
Шаблон:Integers Шаблон:Number theory