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Supersingular Prime (moonshine Theory)
In the mathematical branch of moonshine theory, a supersingular prime is a prime number that divides the order of the Monster group ''M'', which is the largest sporadic simple group. There are precisely fifteen supersingular prime numbers: the first eleven primes ( 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31), as well as 41, 47, 59, and 71. The non-supersingular primes are 37, 43, 53, 61, 67, and any prime number greater than or equal to 73. Supersingular primes are related to the notion of supersingular elliptic curves as follows. For a prime number ''p'', the following are equivalent: # The modular curve ''X''0+(''p'') = ''X''0(''p'') / ''w''p, where ''w''p is the Fricke involution of ''X''0(''p''), has genus zero. # Every supersingular elliptic curve in characteristic ''p'' can be defined over the prime subfield F''p''. # The order of the Monster group is divisible by ''p''. The equivalence is due to Andrew Ogg. More precisely, in 1975 Ogg showed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and mathematical analysis, analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of mathematical object, abstract objects and the use of pure reason to proof (mathematics), prove them. These objects consist of either abstraction (mathematics), abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of inference rule, deductive rules to already established results. These results include previously proved theorems, axioms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
31 (number)
31 (thirty-one) is the natural number following 30 and preceding 32. It is a prime number. In mathematics 31 is the 11th prime number. It is a superprime and a self prime (after 3, 5, and 7), as no integer added up to its base 10 digits results in 31. It is a lucky prime and a happy number; two properties it shares with 13, which is its dual emirp and permutable prime. 31 is also a primorial prime, like its twin prime, 29. 31 is the number of regular polygons with an odd number of sides that are known to be constructible with compass and straightedge, from combinations of known Fermat primes of the form 22''n'' + 1. 31 is the third Mersenne prime of the form 2''n'' − 1. It is also the eighth Mersenne prime exponent, specifically for the number 2,147,483,647, which is the maximum positive value for a 32-bit signed binary integer in computing. After 3, it is the second Mersenne prime not to be a double Mersenne prime. 127, which is the 31st prime number, is a do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fricke Involution
In mathematics, a Fricke involution is the involution of the modular curve ''X''0(''N'') given by τ → –1/''N''τ. It is named after Robert Fricke. The Fricke involution also acts on other objects associated with the modular curve, such as spaces of modular forms and the Jacobian In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to: * Jacobian matrix and determinant * Jacobian elliptic functions * Jacobian variety *Intermediate Jacobian In mathematics, the intermediate Jacobian of a compact Kähle ... ''J''0(''N'') of the modular curve. See also * Atkin–Lehner involution References * Modular forms {{numtheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Curve
In number theory and algebraic geometry, a modular curve ''Y''(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z). The term modular curve can also be used to refer to the compactified modular curves ''X''(Γ) which are compactifications obtained by adding finitely many points (called the cusps of Γ) to this quotient (via an action on the extended complex upper-half plane). The points of a modular curve parametrize isomorphism classes of elliptic curves, together with some additional structure depending on the group Γ. This interpretation allows one to give a purely algebraic definition of modular curves, without reference to complex numbers, and, moreover, prove that modular curves are defined either over the field of rational numbers Q or a cyclotomic field Q(ζ''n''). The latter fact and its general ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supersingular Elliptic Curve
In algebraic geometry, supersingular elliptic curves form a certain class of elliptic curves over a field of characteristic ''p'' > 0 with unusually large endomorphism rings. Elliptic curves over such fields which are not supersingular are called ''ordinary'' and these two classes of elliptic curves behave fundamentally differently in many aspects. discovered supersingular elliptic curves during his work on the Riemann hypothesis for elliptic curves by observing that positive characteristic elliptic curves could have endomorphism rings of unusually large rank 4, and developed their basic theory. The term "supersingular" has nothing to do with singular points of curves, and all supersingular elliptic curves are non-singular. It comes from the phrase "singular values of the j-invariant" used for values of the j-invariant for which a complex elliptic curve has complex multiplication. The complex elliptic curves with complex multiplication are those for which the endomorp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
73 (number)
73 (seventy-three) is the natural number following 72 (number), 72 and preceding 74 (number), 74. In English, it is the smallest natural number with twelve letters in its spelled out name. In mathematics 73 is the 21st prime number, and emirp with 37 (number), 37, the 12th prime number. It is also the eighth twin prime, with 71 (number), 71. It is the largest minimal Primitive root modulo n, primitive root in the first primes; in other words, if ''p'' is one of the first one hundred thousand primes, then at least one of the numbers is a primitive root modulo ''p''. 73 is also the smallest factor of the first Composite number, composite generalized Fermat number in decimal: , and the smallest prime Modular arithmetic#Congruence, congruent to 1 modulo 24 (number), 24, as well as the only prime repunit in base 8 (1118). It is the fourth star number. Notably, 73 is the sole Sheldon prime to contain both ''mirror'' and ''product'' properties: *73, as an emirp, has 37 as its Duali ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
67 (number)
67 (sixty-seven) is the natural number following 66 and preceding 68. It is an odd number. In mathematics 67 is: *the 19th prime number (the next is 71). * a Chen prime. *an irregular prime. *a lucky prime. *the sum of five consecutive primes (7 + 11 + 13 + 17 + 19). *a Heegner number. *a Pillai prime since 18! + 1 is divisible by 67, but 67 is not one more than a multiple of 18. *palindromic in quinary (2325) and senary (1516). *a super-prime. (19 is prime) *an isolated prime. (65 and 69 aren't prime) In science *The atomic number of holmium, a lanthanide. Astronomy *Messier object M67, a magnitude 7.5 open cluster in the constellation Cancer. *The New General Catalogue object NGC 67, an elliptical galaxy in the constellation Andromeda. In music * "Car 67", a song by the band Driver 67 * Chicago's song "Questions 67 and 68" * Elton John's song "Old '67" on '' The Captain & The Kid'' CD, (2006) * British rap group called 67 * Rapper Drake released the song na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
61 (number)
61 (sixty-one) is the natural number following 60 and preceding 62. In mathematics 61 is: *the 18th prime number. *a twin prime with 59. *a cuban prime of the form ''p'' = , where ''x'' = ''y'' + 1. *the smallest ''proper prime'', a prime ''p'' which ends in the digit 1 in base 10 and whose reciprocal in base 10 has a repeating sequence with length ''p'' − 1. In such primes, each digit 0, 1, ..., 9 appears in the repeating sequence the same number of times as does each other digit (namely, times). *the exponent of the 9th Mersenne prime. (261 − 1 = ) *the sum of two squares, 52 + 62. *a centered square number. *a centered hexagonal number. *a centered decagonal number. *the sixth Euler zigzag number (or Up/down number). *a unique prime in base 14, since no other prime has a 6-digit period in base 14. *a Pillai prime since 8! + 1 is divisible by 61 but 61 is not one more than a multiple of 8. *a Keith number, because it recurs in a Fibonacci-like sequence started from its ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
53 (number)
53 (fifty-three) is the natural number following 52 and preceding 54. It is the 16th prime number. In mathematics *Fifty-three is the 16th prime number. It is also an Eisenstein prime, an isolated prime, a balanced prime and a Sophie Germain prime. *The sum of the first 53 primes is 5830, which is divisible by 53, a property shared by only a few other numbers. *In hexadecimal, 53 is 35, that is, the same characters used in the decimal representation, but reversed. Four additional multiples of 53 share this property: 371 = , 5141 = , 99,481 = , and 8,520,280 = 0. Apart from the trivial case of single-digit decimals, no other number has this property. *53 cannot be expressed as the sum of any integer and its decimal digits, making 53 a self number. *53 is the smallest prime number that does not divide the order of any sporadic group. In science *The atomic number of iodine Astronomy * Messier object M53, a magnitude 8.5 globular cluster in the constellation Coma Ber ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
43 (number)
43 (forty-three) is the natural number following 42 and preceding 44. In mathematics Forty-three is the 14th smallest prime number. The previous is forty-one, with which it comprises a twin prime, and the next is forty-seven. 43 is the smallest prime that is not a Chen prime. It is also the third Wagstaff prime. 43 is the fourth term of Sylvester's sequence, one more than the product of the previous terms (2 × 3 × 7). 43 is a centered heptagonal number. Let ''a'' = ''a'' = 1, and thenceforth ''a'' = (''a'' + ''a'' + ... + ''a''). This sequence continues 1, 1, 2, 3, 5, 10, 28, 154... . ''a'' is the first term of this sequence that is not an integer. 43 is a Heegner number. 43 is the largest prime which divides the order of the Janko group J4. 43 is a repdigit in base 6 (111). 43 is the number of triangles inside the Sri Yantra. 43 is the largest natural number that is not an (original) McNugget number. 43 is the smallest prime number expressible as the sum of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
37 (number)
37 (thirty-seven) is the natural number following 36 and preceding 38. In mathematics 37 is the 12th prime number and the third unique prime in decimal. 37 is the first irregular prime, and the third isolated prime without a twin prime. It is also the third cuban prime, the fourth emirp, and the fifth lucky prime. *37 is the third star number and the fourth centered hexagonal number. *The sum of the squares of the first 37 primes is divisible by 37. *Every positive integer is the sum of at most 37 fifth powers (see Waring's problem). *37 appears in the Padovan sequence, preceded by the terms 16, 21, and 28 (it is the sum of the first two of these). *Since the greatest prime factor of 372 + 1 = 1370 is 137, which is substantially more than 37 twice, 37 is a Størmer number. In base-ten, 37 is a permutable prime with 73, which is the 21st prime number. By extension, the mirroring of their digits and prime indexes makes 73 the only Sheldon prime. In moonshine t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
71 (number)
71 (seventy-one) is the natural number following 70 and preceding 72. __TOC__ In mathematics 71 is: *the 20th prime number. The next is 73, with which it composes a twin prime. *a permutable prime and emirp with 17. *is the largest number which occurs as a prime factor of an order of a sporadic simple group. *the sum of three consecutive primes: 19, 23 and 29. *a centered heptagonal number. *an Eisenstein prime with no imaginary part and real part of the form 3''n'' – 1. *a Pillai prime, since 9! + 1 is divisible by 71 but 71 is not one more than a multiple of 9. *the largest (15th) supersingular prime, which is also a Chen prime. *part of the last known pair (71, 7) of Brown numbers, since 712 = 7! + 1. *the twenty-third term of the Euclid–Mullin sequence, as it is the least prime factor of one more than the product of the first twenty-two terms. *the smallest positive integer ''d'' such that the imaginary quadratic fie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
