There are Infinitely Many Fibonacci Primes

We invent a novel algorithm and solve the Fibonacci prime conjecture by an interaction between proof and algorithm. From the entire set of natural numbers successively deleting the residue class 0 mod a prime, we retain this prime and possibly delete another one prime retained, then we invent a recursive sieve method, a modulo algorithm on finite sets of natural numbers, for indices of Fibonacci primes. The sifting process mechanically yields a sequence of sets of natural numbers, which converges to the index set of all Fibonacci primes. The corresponding cardinal sequence is strictly increasing. The algorithm reveals a structure of particular order topology of the index set of all Fibonacci primes, then we readily prove that the index set of all Fibonacci primes is an infinite set based on the existing theory of the structure. Some mysteries of primes are hidden in second order arithmetics.

Multiplication tables for non-prime odd numbers The untouched tables which will give us a better understanding of the distribution of prime odd numbers

The sieve method is used to separate prime numbers from non-prime numbers. If the set of prime odd numbers cannot be written as multiplication tables, the set of non-prime odd numbers can be written as multiplication tables. Thus, each odd number that does not appear in these multiplication tables is certainly a prime odd number. Based on these tables, it was proved by the opposite method that the series of prime numbers are random series. Although they are random, they can be easily tracked using the opposite method. The counter-example is used to proof that it is not possible to write whole multiplication tables of prime odd numbers on formula of [(𝑎×𝑏)+𝑐] or [(𝑎×𝑏)−𝑐]. Instead, partial multiplication tables can be used. It also was proved that the number 1 is a prime odd number.

Counting the number of trigonal curves of genus 5 over finite fields

The trigonal curves of genus 5 can be represented by projective plane quintics that have one singularity of delta invariant one. Combining this with a partial sieve method for plane curves we count the number of such curves over any finite field. The main application is that this gives the motivic Euler characteristic of the moduli space of trigonal curves of genus 5.

Performance of Mastication in Menopausal Women in Palembang

Mastication is the mechanical process of breaking down food in oral cavity. The mastication system involves several components, including saliva and alveolar bone. Salivary flow rate and alveolar bone mineral density have been reported decreased in menopausal women due to the impact of declining estrogen levels. Menopause is the permanent cessation of menstruation resulting from the loss of ovarian follicular activity. The aim of this study was to evaluate the masticatory performance in menopausal women in Kelurahan Talang Kelapa RW 19. This observational experimental study with cross-sectional design involved 36 women, consisting of 18 women and 18 non-menopause women. Masticatory performance was evaluated using a sieve method. Each subject was asked to chew 3 gram peanuts for 20 masticatory strokes, which that procedure was repeated for five times. Masticatory performance was measured using a sieve method which based on the assessment of the size of food particles that have been chewed and filtered over a number 10 mesh sieve. Percentage masticatory performance was obtained by dividing volume of particles passed through the filter with the volume of total particles. The masticatory performance data were analyzed statically with independent samples T test. The mean masticatory performance score was 13,71%, in menopausal women and 30,62% in non-menopausal women. It was found that masticatory performance in menopausal women was significantly lower when compared to non-menopausal women (p0,05). It may be concluded that menopause decreased the masticatory performance.

Large prime factors on short intervals

We show that for all large enough x the interval [x, x + x1/2 log1.39x] contains numbers with a prime factor p > x18/19. Our work builds on the previous works of Heath–Brown and Jia (1998) and Jia and Liu (2000) concerning the same problem for the longer intervals [x, x + x1/2 + ϵ]. We also incorporate some ideas from Harman’s book Prime-detecting sieves (2007). The main new ingredient that we use is the iterative argument of Matomäki and Radziwiłł (2016) for bounding Dirichlet polynomial mean values, which is applied to obtain Type II information. This allows us to take shorter intervals than in the above-mentioned previous works. We have also had to develop ideas to avoid losing any powers of log x when applying Harman’s sieve method.