Intersective polynomials and the primes

Intersective polynomials are polynomials in $\Z[x]$ having roots every modulus. For example, $P_1(n)=n^2$ and $P_2(n)=n^2-1$ are intersective polynomials, but $P_3(n)=n^2+1$ is not. The purpose of this note is to deduce, using results of Green-Tao \cite{gt-chen} and Lucier \cite{lucier}, that for any intersective polynomial $h$, inside any subset of positive relative density of the primes, we can find distinct primes $p_1, p_2$ such that $p_1-p_2=h(n)$ for some integer $n$. Such a conclusion also holds in the Chen primes (where by a Chen prime we mean a prime number $p$ such that $p+2$ is the product of at most 2 primes)

The effects of bismuth, strontium and antimony additions on the microstructure and mechanical properties of A356 aluminium casting alloy

Aluminium castings offer significant weight reduction and improved fuel efficiency. Nowadays, aluminium recycling is widely practiced so impurity-related problems has become more important. Bismuth is one of the alloying elements added to aluminium alloys to improve their machinability, but little is known about its effect as a modifier or refiner. There has also been little investigation on the effect of low strontium contents (0.001wt% to 0.006wt%) on porosity formation. In the present work both sand and permanent moulds were used to produce bars containing varying strontium-bismuth ratios with some being treated with 0.2wt% antimony to investigate the interaction between these elements. A quench-during-solidification technique had been performed to study the effect of low strontium content on nucleation and growth of porosity in A356 alloy. Optical microscope, image analyzer, scanning electron microscopy (SEM), energy dispersive x-ray analysis (EDX) and x-ray diffraction (XRD) analysis were used to characterize the eutectic silicon, porosity and other phases. Strontium content as low as 0.004wt% was found to bring upon modification to the morphology of the eutectic silicon, whereas an addition of 0.005wt% bismuth refined the eutectic silicon. Beyond this level of bismuth the silicon phase was found to undergo coarsening. A strontium-bismuth ratio of at least 0.5 is suggested to be necessary to ensure a modified silicon morphology, whereas the refining effect of antimony was not affected by bismuth addition. Percentage area of porosity and pore roundness were found to increase with increasing strontium content, reasonably due to earlier pore growth and less shrinkage-type porosity in the castings. The nucleation of new pores occurred at the solid fraction of around 75%, regardless of strontium content. In the present work, the effect of low strontium content, cooling rate and heat treatment (T6) on the mechanical properties was also studied. The results showed that the mechanical properties were less affected by the strontium level but more by heat treatment and cooling rates

Readiness to teach secondary mathematics: A study of pre-service mathematics teachers\u27 self-perceptions

This study evaluated pre-service teachers’ (PSMTs) perceptions of their own readiness to teach secondary mathematics. The study was conducted at an Australian university across two campuses, in different states. Specifically, PSMTs’ perceptions of their preparedness were explored in terms of mathematical content knowledge, pedagogical content knowledge, and mathematical knowledge for teaching. Findings indicate that while the majority of participants feel that they have the requisite content knowledge to confidently teach Lower School secondary mathematics, further training is required to develop their content and pedagogical knowledge, especially for upper secondary mathematics