INTEGER STRUCTURE ANALYSIS OF PRIMES AND COMPOSITES FROM SUMS OF TWO FOURTH POWERS

Abstract An integer structure (IS) analysis of the sum ( ) Z . This sum generates many primes and the row structure of such primes is explored. The class functions of the composite factors of this sum are also given, and these, together with the associated row functions, illustrate why it is impossible to produce an integer to the fourth power from such sums. The overall results are consistent with those previously found with IS analysis

The Pascal-Fibonacci numbers

Abstract. The Pascal-Fibonacci (PF) numbers for a given Fibonacci number sum to give the values of that Fibonacci number. Individual PF numbers are members of one of the triangular, tetrahedral or pentagonal series or have factors in common with the pyramidal or other geometric series. For composite numbers, partial sums of PF numbers can yield a factor, while prime Fibonacci numbers are detected with sums of squares

Heat and mass transfer effects of ice growth mechanisms in pure water and aqueous solutions

Interactions between heat and mass diffusion determine growth mechanisms during ice crystallization. The effects of heat and mass transfer on ice growth in pure water and magnesium sulfate solution were investigated by studying the evolution of the gradient of the refractive index using color Schlieren deflectometry. For pure water, the gradient of the refractive index of water was used to calculate the temperature and therefore the local supersaturation. Its effect on the ice crystal growth rate and morphology was studied. It was found that, for local supersaturations greater than 2.8, the morphology was dendritic ice, with a growth rate 2 orders of magnitude higher than that for layered growth. During dendritic growth, 3−16% of the heat of crystallization diffused to the liquid side, which is counter to current understanding. At the transition (between the time of partial melting of the dendritic ice and the beginning of the layered ice growth), a higher supersaturation than that responsible for layered growth was observed. For ice growth from an aqueous salt solution, a mass and thermal diffusion boundary layer in front of the growing ice was created by diffusion of the solutes from the ice and by the release of heat of crystallization