Proof the Skewes’ number is not an integer using lattice points and tangent line

Skewes’ number was discovered in 1933 by South African mathematician Stanley Skewes as upper bound for the first sign change of the difference π (x) − li(x). Whether a Skewes’ number is an integer is an open problem of Number Theory. Assuming Schanuel’s conjecture, it can be shown that Skewes’ number is transcendental. In our paper we have chosen a different approach to prove Skewes’ number is an integer, using lattice points and tangent line. In the paper we acquaint the reader also with prime numbers and their use in RSA coding, we present the primary algorithms Lehmann test and Rabin-Miller test for determining the prime numbers, we introduce the Prime Number Theorem and define the prime-counting function and logarithmic integral function and show their relation.

Changes in Abrasive Wear Resistance during Miller Test of High-Manganese Cast Steel with Niobium Carbides Formed in the Alloy Matrix

High-manganese Hadfield cast steel is commonly used for machine components operating under dynamic load conditions. The high fracture toughness and abrasive wear resistance of this steel are the result of an austenitic structure, which—while being ductile—at the same time tends to surface harden under the effect of cold work. Absence of dynamic loads (e.g., in the case of sand abrasion) causes rapid and premature wear of parts. To improve the abrasive wear resistance of high-manganese cast steel for operation under the conditions free from dynamic loads, primary niobium carbides are produced in this cast steel during the melting process to obtain in castings, after melt solidification, the microstructure consisting of an austenitic matrix and primary niobium carbides uniformly distributed in this matrix. The measured hardness of the tested samples as cast and after solution heat treatment is 260–290 HV and is about 30–60 HV higher than the hardness of common Hadfield cast steel, which is 230 HV. Compared to common Hadfield cast steel, the abrasive wear resistance of the tested high-manganese cast steel measured in the Miller test is at least three times higher at the niobium content of 3.5 wt%. Increasing the niobium content to 4.5 wt%. in the tested samples increases this wear resistance even more.

High particle variability across siliconized and oil-free syringes and needles from the same lots

Previous studies have reported silicone oil (SO) applied to needles and syringes in the vitreous of patients after intravitreal injections. We evaluated four syringes (SR 1-mL insulin, Saldanha-Rodrigues; BD 1-mL Tuberculin Slip Tip, Becton–Dickinson; BD Ultra-Fine 0.3 mL, HSW Norm-Ject Tuberculin, Henke Sass Wolf) and 10 needles (BD PrecisionGlide 27- and 30-gauge (G); BD Eclipse and JBP Nanoneedle 27-, 30-, 33- and 34-G; TSK Invisible Needle and 27 and 30-G Steriject Control Hub). The protein-free buffer samples injected into the syringes and needles under study were collected in an Eppendorf tube and taken to Flow imaging microscopy, that characterized the concentration and morphology of the microsized particles. The number of particles was analyzed. The coefficients of variation (CV) were the primary outcome. The Feltz and Miller test compared the CVs. The significance level was 5%. Numerous particles and high CVs were associated with both devices, needles and syringes; the comparisons among them did not reach significance. The BD Ultrafine 0.3 mL syringe (149.7%) had the highest CV and the SO-free HSW Norm-Ject (66.4%) syringe the lowest, and the TSK Invisible needle (149.5%) had the highest and the BD Precision Glide 30G needle (35.9%) needle the lowest. In conclusion, particle release, including those with SO morphology, varied greatly among instruments, even from the same lots, which is relevant considering that fewer particles are injected into some eyes compared with others.

Autistic Savants and large number primality detection.

In a sequel to the paper on small number primality detection by mental arithmetic.In this paper, we consider primality detection of four digit prime numbers, leading next to larger six digit and eight digit numbers, optionally scaled to arbitrary sized numbers. Several mental arithmetic techniques as mental arithmetic exercises from literature are cited, towards effective primality detection, by mental arithmetic only.The “large primes” mental arithmetic skill is developed both as a web based UI and as a UI based on slack, using the Wolfram Alpha nd Wolfram API , for primality testing and for Easter Eggs on prime numbers.Keywords: ASD, prime number determination, autistic savants, mathematical testing, Alexa Skills, learning and Cognition, Rabin-Miller test, Lucas test, fermat primes.

Evaluation of surface roughness and color change of bovine enamel after immersion in dye solution

Aim: The aim of this in vitro study was to evaluate the influence of dye solution on enamel color change after bleaching protocols and the effectiveness in maintaining the color of these agents.Material and Methods: The buccal surfaces of sixty-five bovine incisors were cleaned and polished, and the enamel specimens were divided into thirteen groups: G1 to G6: treated with 6% hydrogen peroxide using different surface agents; G7 to G12: treated with 15% hydrogen peroxide using different surface agents; G13: control. After 24 hours, the groups treated were immersed in black tea solution; the control group was stored in artificial saliva. The color was evaluated prior to bleaching, 24 hours later and after immersion in the dye solution; the roughness was measured immediately after bleaching, 24 hours later and 7 days after immersion in the dye solution. The data was analyzed using the Kruskal-Wallis test, followed by the Miller test for roughness analysis, and the Duncan test for color change analysis. It was used 5% significant level with p<0.05.Results: The data found in the evaluation of surface roughness after bleaching indicated a reduction of roughness in all the groups. The surface agent Bifluoride, when applied, showed an increase in roughness after its application and it decreases after immersion in dye solution; the surface agent Desensibilize and the XP Bond adhesive showed greater color alteration after immersion in dye solution.Conclusions: All the groups studied, under different whitening technique, were effective in promoting whitening.

Effects of Rare Earth Metal Addition on Wear Resistance of Chromium-Molybdenum Cast Steel

This paper discusses changes in the microstructure and abrasive wear resistance of G17CrMo5-5 cast steel modified with rare earth metals (REM). The changes were assessed using scanning microscopy. The wear response was determined in the Miller test to ASTM G75. Abrasion tests were supplemented with the surface profile measurements of non-modified and modified cast steel using a Talysurf CCI optical profilometer. It was demonstrated that the modification substantially affected the microstructure of the alloy, leading to grain size reduction and changed morphology of non-metallic inclusions. The observed changes in the microstructure resulted in a three times higher impact strength (from 33 to 99 kJ/cm2) and more than two times higher resistance to cracking (from 116 to 250 MPa). The following surface parameters were computed: Sa: Arithmetic mean deviation of the surface, Sq: Root-mean-square deviation of the surface, Sp: Maximum height of the peak Sv: Maximum depth of the valley, Sz: Ten Point Average, Ssk: Asymmetry of the surface, Sku: Kurtosis of the surface. The findings also indicated that the addition of rare earth metals had a positive effect on the abrasion behaviour of G17CrMo5-5 cast steel.

Wear Resistance of 18%Cr-9%Ni Steel Used for Cast Parts of Pumps Operating in Corrosive – Erosive Environments

This paper presents the results of experimental studies, the main aim of which has been to demonstrate that changes in the microstructure of austenitic 18%Cr-9%Ni cast steel provoked by the addition of 1.4% boron, and boron with titanium, give increased wear resistance. After melting the high-alloyed 18%Cr-9%Ni cast steel with an addition of boron, and boron with titanium, metallographic examinations were conducted using light microscopy and SEM. These examinations revealed in the austenitic structure of the 18%Cr-9%Ni cast steel, the presence of a eutectic rich in boron and chromium, and characterised by a microhardness of 1838-1890 μ HV20. Additionally, in the cast steel inoculated with boron and titanium, the presence of titanium nitride precipitates was observed. Changes that have occurred in the microstructure as a result of introducing the additions of boron, and boron with titanium, also caused an increase of the cast steel hardness from 212 HV30 to 290-320 HV30 and 320-350 HV30, respectively. To determine the abrasive wear resistance, 16-hour Miller test was performed (ASTM G 75-07), wherein the abrasive medium was a mixture of SiC and water. Obtaining the hard, rich in boron and chromium, eutectic and titanium nitride precipitates in the structure of 18%Cr-9%Ni cast steel increased the abrasive wear resistance by approximately 21%, according to the data recorded in the sixteenth hour of the test cycle. As an additional benchmark point for the results obtained served the wear resistant, structural, L35GSM steel used for castings working in difficult conditions. Comparing the values of abrasive wear resistance obtained for the 18%Cr-9%Ni cast steel and cast L35GSM steel, an increase in the wear resistance of the 18%Cr-9%Ni cast steel by about 35% has been proved.