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The congruence relation, modulo m, partitions the set of integers into m congruence classes. Operations of addition and multiplication can be defined on these m objects in the following way: To either add or multiply two congruence classes, first pick a representative (in any way) from each class, then perform the usual operation for integers on the two representatives and finally take the ...
In mathematics, a Ramanujan–Sato series [1] [2] generalizes Ramanujan’s pi formulas such as, = = ()!! + to the form = = + by using other well-defined sequences of integers obeying a certain recurrence relation, sequences which may be expressed in terms of binomial coefficients (), and ,, employing modular forms of higher levels.
It can be shown that Δ is a modular form of weight twelve, and g 2 one of weight four, so that its third power is also of weight twelve. Thus their quotient, and therefore j , is a modular function of weight zero, in particular a holomorphic function H → C invariant under the action of SL(2, Z ) .
When R is a power of a small positive integer b, N′ can be computed by Hensel's lemma: The inverse of N modulo b is computed by a naïve algorithm (for instance, if b = 2 then the inverse is 1), and Hensel's lemma is used repeatedly to find the inverse modulo higher and higher powers of b, stopping when the inverse modulo R is known; N′ is ...
The modular multiplicative inverse is defined by the following rules: Existence: There exists an integer denoted a −1 such that aa −1 ≡ 1 (mod m) if and only if a is coprime with m. This integer a −1 is called a modular multiplicative inverse of a modulo m.
L = p 1 p 2 …p m = q 1 q 2 …q n . Since each prime p divides L by assumption, it must also divide one of the q factors; since each q is prime as well, it must be that p = q. Iteratively dividing by the p factors shows that each p has an equal counterpart q; the two prime factorizations are identical except for their order. The unique ...
5.3 The constants e 1, e 2 and e 3. ... The series expansion suggests that g 2 and g 3 are homogeneous functions of degree − ... The discriminant is a modular form ...
The ring of level 1 modular forms is generated by the Eisenstein series E 4 and E 6 (which generate the ring of holomorphic modular forms) together with the inverse 1/Δ of the modular discriminant. Any weakly holomorphic modular form of any level can be written as a quotient of two holomorphic modular forms.