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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 \mathbb_p. # The order of the Monster group is divisible by p. The equivalence is due to Andrew Ogg. More precisely, in 1975 Ogg showed that the primes satisfying the first condition ... [...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 -invariant for which a complex elliptic curve has complex multiplication. The complex elliptic curves with complex multiplication are those for which the endomorphism ring has t ... [...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 ''J''0(''N'') of the modular curve. The quotient of ''X''0(''N'') by the Fricke involution is a curve called ''X''0+(''N''), and for ''N'' prime this has genus zero only for a finite list of primes, called supersingular primes, which are the primes that divide the order of the Monster group In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group; it has order : : = 2463205976112133171923293 .... See also * Atkin–Lehner involution References * Modular forms {{numtheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
31 (number)
31 (thirty-one) is the natural number following thirty, 30 and preceding 32 (number), 32. It is a prime number. Mathematics 31 is the 11th prime number. It is a superprime and a Self number#Self primes, self prime (after 3, 5, and 7), as no integer added up to its base 10 digits results in 31. It is the third Mersenne prime of the form 2''n'' − 1, and the eighth Mersenne prime ''exponent'', in-turn yielding the maximum positive value for a 32-bit Integer (computer science), signed binary integer in computing: 2,147,483,647. After 3, it is the second Mersenne prime not to be a double Mersenne prime, while the 31st prime number (127 (number), 127) is the second double Mersenne prime, following 7. On the other hand, the thirty-first triangular number is the perfect number 496 (number), 496, of the form 2(5 − 1)(25 − 1) by the Euclid-Euler theorem. 31 is also a ''primorial prime'' like its twin prime (29 (number), 29), as well as both a lucky prime and a happy number like its d ... [...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 generaliz ... [...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. It is the 21st prime number and the fourth star number. It is also the eighth twin prime, with 71 (number), 71. 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 100,000 primes; in other words, if is one of the first one hundred thousand primes, then at least one of the numbers 2, 3, 4, 5, 6, ..., 73 is a primitive root modulo . 73 is also the smallest factor of the first Composite number, composite generalized Fermat number in decimal: 10^+1=10,001=73\times 137, and the smallest prime Modular arithmetic#Congruence, congruent to 1 modulo 24 (number), 24, as well as the only prime repunit in octal (1118). It is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
67 (number)
67 (sixty-seven) is the natural number following 66 (number), 66 and preceding 68 (number), 68. It is an Parity (mathematics), odd and prime number. In mathematics 67 is: *the 19th prime number (the next is 71 (number), 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 Twin prime#Isolated prime, isolated prime. (65 and 69 are not prime) *a sexy prime with 61 and 73 References External links Integers {{Num-stub ... [...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, and a twin prime with 59. As a centered square number, it is the sum of two consecutive squares, 5^2 + 6^2. It is also a centered decagonal number, and a centered hexagonal number. 61 is the fourth cuban prime of the form p = \frac where x = y + 1, and the fourth Pillai prime since 8! + 1 is divisible by 61, but 61 is not one more than a multiple of 8. It is also a Keith number, as it recurs in a Fibonacci-like sequence started from its base 10 digits: 6, 1, 7, 8, 15, 23, 38, 61, ... 61 is a unique prime in base 14, since no other prime has a 6-digit period in base 14, and palindromic in bases 6 (1416) and 60 (1160). It is the sixth up/down or Euler zigzag number. 61 is the smallest ''proper prime'', a prime p which ends in the digit 1 in decimal and whose reciprocal in base-10 has a repeating sequence of length p - 1, where each digit (0, 1, ..., 9) ap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
53 (number)
53 (fifty-three) is the natural number following 52 (number), 52 and preceding 54 (number), 54. It is the 16th prime number. In mathematics Fifty-three is the 16th prime number. It is the second balanced prime, and fifth isolated prime. 53 is a sexy prime with 47 (number), 47 and 59 (number), 59. It is the eighth Sophie Germain prime, and the ninth Eisenstein prime. The sum of the first 53 primes is 5830, which is divisible by 53, a property shared by only a few other numbers. 53 cannot be expressed as the sum of any integer and its decimal digits, making 53 the ninth self number in decimal. 53 is the smallest prime number that does not divide the order of any sporadic group, inclusive of the six Pariah group, pariahs; it is also the first prime number that is not a member of Bhargava's prime-universality criterion theorem (followed by the next prime number 59 (number), 59), an integer-matrix quadratic form that represents all prime numbers when it represents the sequence of s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
43 (number)
43 (forty-three) is the natural number following 42 (number), 42 and preceding 44 (number), 44. Mathematics 43 is a prime number, and a twin prime of 41 (number), 41. 43 is the smallest prime that is not a Chen prime. 43 is also a Wagstaff prime, and a Heegner number. 43 is the fourth term of Sylvester's sequence. 43 is the largest prime which divides the order of the Janko group J4, Janko group J4. Netherlands, Dutch mathematician Hendrik Lenstra wrote a mathematical research paper discussing the properties of the number, titled ''Ode to the number 43.'' Notes Further reading Hendrik Lenstra, Lenstra, Hendrik (2009)''Ode to the number 43'' (In Dutch). Nieuw Archief voor Wiskunde, Nieuw Arch. Wiskd. Amsterdam, NL: Koninklijk Wiskundig Genootschap (5) 10, No. 4: 240-244. {{Integers, zero Integers ... [...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 3rd isolated prime without a twin prime. 37 is the first irregular prime with irregularity index of 1, where the smallest prime number with an irregularity index of 2 is the thirty-seventh prime number, 157. The smallest magic square, using only primes and 1, contains 37 as the value of its central cell: Its magic constant is 37 x 3 = 111, where 3 and 37 are the first and third base-ten unique primes (the second such prime is 11). 37 requires twenty-one steps to return to 1 in the Collatz problem, as do adjacent numbers 36 and 38. The two closest numbers to cycle through the elementary Collatz pathway are 5 and 32, whose sum is 37; also, the trajectories for 3 and 21 both require seven steps to reach 1. On the other hand, the first two integers that return 0 for the Mertens function ( 2 and 39) have a difference of 37, where the ... [...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. Because both rearrangements of its digits (17 and 71) are prime numbers, 71 is an emirp and more generally a permutable prime. 71 is a centered heptagonal number. It is a regular prime, a Ramanujan prime, a Higgs prime, and a good prime. It is a Pillai prime, since 9!+1 is divisible by 71, but 71 is not one more than a multiple of 9. It is part of the last known pair (71, 7) of Brown numbers, since 71^=7!+1. 71 is the smallest of thirty-one discriminants of imaginary quadratic fields with class number of 7, negated (see also, Heegner number In number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from int ...s). 71 is the largest number which occurs as a prim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
