Number-theoretic expressions obtained through analogy between prime factorization and optical interferometry
نویسندگان
چکیده
منابع مشابه
Applications of Prime Factorization of Ideals in Number Fields
For a number fieldK, that is, a finite extension of Q, and a prime number p, a fundamental theorem of algebraic number theory implies that the ideal (p) ⊆ OK factors uniquely into prime ideals as (p) = p1 1 · · · p eg g . In this paper we explore different interpretations of this using the factorization of polynomials in finite and p-adic fields and Galois theory. In particular, we present some...
متن کاملFactorization of Prime Ideal Extensions in Number Rings
Following an idea of Kronecker, we describe a method for factoring prime ideal extensions in number rings. The method needs factorization of polynomials in many variables over finite fields, but it works for any prime and any number field extension. Introduction Let F c K be number fields, let ¿fp c <?* be their corresponding number rings, i.e., the integral closures of Z in F and K, respective...
متن کاملNumber theoretic transform modulo K.2N+1, a prime
Due to its simple and real arithmetic structure Number Theoretic Transform is attractive for computation of convolution. However, there exists a stringent relation between the choice of modulus M and convolution length. Choice of modulus as K.2+1, a prime, leads to relaxation of this constraint and wide choices of wordlength, with each of these associated with many choices of convolution length...
متن کاملThe analogy between optical beam shifts and quantum weak measurements
We describe how the notion of optical beam shifts (including the spatial and angular Goos–Hänchen shift and Imbert–Federov shift) can be understood as a classical analogue of a quantum measurement of the polarization state of a paraxial beam by its transverse amplitude distribution. Under this scheme, complex quantum weak values are interpreted as spatial and angular shifts of polarized scalar ...
متن کاملIntegers, Prime Factorization, and More on Primes
The integer q is called the quotient and r is the remainder. Proof. Consider the rational number b a . Since R = ⋃ k∈Z[k, k + 1) (disjoint), there exists a unique integer q such that b a ∈ [q, q + 1), i.e., q ≤ b a < q + 1. Multiplying through by the positive integer a, we obtain qa ≤ b < (q + 1)a. Let r = b− qa. Then we have b = qa + r and 0 ≤ r < a, as required. Proposition 3. Let a, b, d ∈ Z...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Physical Review A
سال: 2012
ISSN: 1050-2947,1094-1622
DOI: 10.1103/physreva.85.043842