A prime number p is called elite if only finitely many Fermat numbers 2 n + 1 are quadratic residues of p. Previously only the interval up to 10 was systematically searched for elite primes and 16 such primes were found. We extended this research up to 2.5 · 10 and found five further elites, among which 1 151 139 841 is the smallest and 171 727 482 881 the largest.

In the book which accompanies the stencils various methods of finding quadratic residues are discussed and the method by means of the expansion of the square root of the number in a continued fraction is found to be by far the most effective. Various examples are given illustrating the power of the stencils, and a reproduction of the first page of the list of primes accompanies the work. The pl...

C4ST-1 (chondroitin 4-sulphotransferase-1) and C6ST-1 (chondroitin 6-sulphotransferase-1) transfer sulphate from PAPS (adenosine 3'-phosphate 5'-phosphosulphate) to positions 4 and 6 respectively of the GalNAc residues of chondroitin. We showed previously that C4ST-1 purified from rat chondrosarcoma and recombinant C4ST-1 both transfer sulphate efficiently to position 4 of the GalNAc residues o...

This paper describes techniques to automatically morph from one sound to another. Audio morphing is accomplished by representing the sound in a multi-dimensional space that is warped or modified to produce a desired result. The multi-dimensional space encodes the spectral shape and pitch on orthogonal axes. After matching components of the sound, a morph smoothly interpolates the amplitudes to ...

In this manuscript novel architectures for modulo 2 þ 1 multi-operand addition and residue generation are introduced. The proposed arithmetic components consist of a translation stage, an inverted endaround-carry carry-save-adder tree and an enhanced diminished-1 modulo 2 þ 1 adder. Qualitative and quantitative results indicate that the proposed architectures result in significantly faster and ...

Despite difficulties in general division, magnitude comparison, and sign detection, residue number system arithmetic has been used for many special-purpose systems in light of its parallelism and modularity for the most common arithmetic operations of addition/subtraction and multiplication. Computation in RNS requires modular reduction, both for the initial conversion from binary to RNS and af...

In [1] a proof was given of Fermat’s Two-Square Theorem using the group theoretical structure of the classical modular group. This has been extended in many directions and to other square properties in general rings. In particular in [2] a two-square theorem was given for the Gaussian integers in terms of when ii is a quadratic residue. In this note we examine and survey this technique and the ...