m = 0: When m = 0, Z/0 is Z, because every number is equivalent to itself (mod 0), but not to any other. But Z does not have multiplicative inverses, and so it cannot be a field. m = 1: When m = 1, Z/Z is {0}, because every number is equivalent to every other number (mod 1). But {0} does not have a non-zero multiplicative inverse, because 1 ≡ 0, and so {0} is not a field. m ≥ 4: Here, we have t...

Introduction – Study of an Elementary Proof 1 1. The Lemmas of Thue and Vinogradov 4 2. Preliminaries on Quadratic Reciprocity and Quadratic Forms 4 2.1. Quadratic reciprocity law 4 2.2. Binary quadratic forms 5 3. Thue-Vinogradov Applied to Binary Quadratic Forms 7 4. First Applications of Theorem 9 8 4.1. Indefinite forms 8 4.2. Positive definite forms 10 5. Primes of the form x +Dy for idone...

A 4 copies of the manuscript A please indicate the corresponding author's details on the manuscript (tel/fax/e-mail) A 4 sets of ®gures A electronic version of the manuscript (Label the ¯oppy disk with the type of computer and word processing package used, your name and the name of the ®le on the disk.) A electronic version of the ®gures (optional, see instructions below) For Review Papers: A 3...

Fix a prime/?. We say that a set S forms a complete residue system modulop if, for all i such that 0 < i < p 1 , there exists s GS such that s = J (mod /?). We say that a set S forms a reduced residue system modulo p if, for all / such that 1 < i < p -1, there exists s GS such that s = i (mod p). In [9], Shah showed that, ifp is a prime and p = 1,9 (mod 10), then the Fibonacci sequence does not...

One says g is a Fibonacci primitive root modulo /?, wherep is a prime, iff g is a primitive root modulo/7 and g = g + 1 (mod p). In [1 ] , [2 ] , and [3] some interesting properties of Fibonacci primitive roots were developed. In this paper, we shall show that a necessary and sufficient condition for a prime/? ^ 5 to have a Fibonacci primitive root is p = 1 or 9 (mod 10) and Alp) = p 1, where/I...

We determined the Newtonian constant of gravitation G by interferometrically measuring the change in spacing between two free-hanging pendulum masses caused by the gravitational field from large tungsten source masses. We find a value for G of (6.672 34±0.000 14)×10(-11)  m3  kg(-1)  s(-2). This value is in good agreement with the 1986 Committee on Data for Science and Technology (CODATA) value...