Exponential sums over Mersenne numbers
نویسندگان
چکیده
منابع مشابه
Sums of Prime Divisors and Mersenne Numbers
The study of the function β(n) originated in the paper of Nelson, Penney, and Pomerance [7], where the question was raised as to whether the set of Ruth-Aaron numbers (i.e., natural numbers n for which β(n) = β(n+ 1)) has zero density in the set of all positive integers. This question was answered in the affirmative by Erdős and Pomerance [5], and the main result of [5] was later improved by Po...
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A fundamental question (perhaps the fundamental question) we can ask about such an ƒ is: Does ƒ = 0 have a solution in integers? Clearly, if there exists an integer solution, then for any prime p there exists a solution in the finite field Z/pZ ; simply reduce mod p any integer solution. (For any ring R, ƒ maps R to R. When we say "a solution of ƒ = 0 in R," we mean an «-tuple of elements (a{, ...
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Generalised Mersenne Numbers (GMNs) were defined by Solinas in 1999 and feature in the NIST (FIPS 186-2) and SECG standards for use in elliptic curve cryptography. Their form is such that modular reduction is extremely efficient, thus making them an attractive choice for modular multiplication implementation. However, the issue of residue multiplication efficiency seems to have been overlooked....
متن کاملMersenne and Fermat Numbers
The first seventeen even perfect numbers are therefore obtained by substituting these values of ra in the expression 2n_1(2n —1). The first twelve of the Mersenne primes have been known since 1914; the twelfth, 2127 —1, was indeed found by Lucas as early as 1876, and for the next seventy-five years was the largest known prime. More details on the history of the Mersenne numbers may be found in ...
متن کاملExponential Sums over Points of Elliptic Curves
We derive a new bound for some bilinear sums over points of an elliptic curve over a finite field. We use this bound to improve a series of previous results on various exponential sums and some arithmetic problems involving points on elliptic curves. Subject Classification (2010) Primary 11L07, 11T23 Secondary 11G20
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ژورنال
عنوان ژورنال: Compositio Mathematica
سال: 2004
ISSN: 0010-437X,1570-5846
DOI: 10.1112/s0010437x03000022