scholarly journals On the greatest prime Factor of a polynomial

1969 ◽  
Vol 16 (4) ◽  
pp. 301-303 ◽  
Author(s):  
M. Keates
Keyword(s):  
Positive Constant ◽  
Prime Factor ◽  
Rational Integer ◽  
Integer Coefficients ◽  
Special Polynomials ◽  
Linear Polynomial ◽  
Greatest Prime Factor ◽  
Absolute Positive Constant ◽  
Non Linear

Let ƒ(x) be a non-linear polynomial with rational integer coefficients, and for integral x let P(x) denote the greatest (positive) prime factor of ƒ(x). Pólya (1) has proved that if ƒ(x) is of degree 2 and has distinct roots then P(x)→∞ as x→∞. It is probably well-known that, provided ƒ(x) has distinct roots, this is true whatever the degree of ƒ(x). There does not appear to be a proof of this in the literature, but it is easily deducible from a result of Siegel (2). These results, however, are non-effective, although effective results have been obtained for a number of special polynomials. Chowla (3) has proved that, if ƒ(x) = x2 +1, then P(x)>C log log x, where C is an absolute positive constant. Analogous results have been proved for some polynomials of the form ax2+b and for some of the form ax3 + b by Mahler and Nagell respectively (4).

Extension of Parity Space to Non Linear Polynomial Dynamic Systems

1997 ◽  
Vol 30 (18) ◽  
pp. 857-862 ◽  
Author(s):  
C. Guernez ◽  
J.Ph. Cassar ◽  
M. Staroswiecki
Keyword(s):  
Dynamic Systems ◽  
Linear Polynomial ◽  
Non Linear ◽  
Parity Space

scholarly journals Perbandingan Forecasting Metode Regresi Non-Linear Polinomial dengan Logika Fuzzy Pada Pemetaan Potensi Bisnis Lampu Berbasis Reduse, Reuse dan Recycle

2018 ◽  
Vol 1 (2) ◽  
pp. 71
Author(s):  
Sari Ayu Wulandari ◽  
Wisnu Adi Prasetyanto ◽  
Rudy Tjahyono
Keyword(s):  
Linear Polynomial ◽  
Non Linear

Mitra pada program PKM ini adalah BSRB (Bank Sampah Resik Becik) di Kota Semarang. Permasalahan yang dihadapi oleh mitra PKM adalah adanya penimbunan sampah bohlam lampu, hal ini karena mereka belum dapat mampu melakukan reuse untuk jenis sampah ini. Sebenarnya reuse bulb (rebu), sangatlah mudah dan dapat dilakukan oleh siapapun. Dengan permasalahan yang dialami oleh mitra, maka sangat diperlukan sebuah teknologi terapan dari proses reuse dari sampah bohlam lampu, yang diwujudkan dalam workshop pelatihan pengolahan hingga proses design product akhir dan penjualan. Penerapan teknologi ini diharapkan dapat meningkatkan produktifitas dan produk hasil bagi mitra UMKM BSRB di Kota Semarang. Dengan mengaplikasikan teknologi rebu (reuse bulb) yang menggunakan Bulb bekas ini adalah meningkatkan produktifitas mitra. Masyarakat wajib mengetahui nilai ekonomis produk, agar dapat melihat peluang bisnis. Peluang bisnis dirangkum dalam data yang akan di forecasting untuk kebutuhan tertentu. Pada artikel pengabdian masyarakat kali ini, akan dibandingkan forecasting antara penggunaan regresi non linear polynomial dengan metode logika fuzzy, yaitu menggunakan fuzzy mamdani. Dengan mengetahui prospek bisnis beberapa tahun berikutnya, masyarakat binaan dapat mempersiapkan dan semangat dalam melaksanakan kegiatan reduce, reuse dan recycle.

On Functional Properties of Incomplete Gaussian Sums

1991 ◽  
Vol 43 (1) ◽  
pp. 182-212 ◽  
Author(s):  
K. I. Oskolkov
Keyword(s):  
Functional Properties ◽  
Positive Constant ◽  
Absolute Constant ◽  
Special Function ◽  
Absolute Positive Constant ◽  
Image Position ◽  
Gaussian Sums

AbstractThe following special function of two real variables x2 and x1 is considered: and its connections with the incomplete Gaussian sums where ω are intervals of length |ω| ≤1. In particular, it is proved that for each fixed x2 and uniformly in X2 the function H(x2, x1) is of weakly bounded 2-variation in the variable x1 over the period [0, 1]. In terms of the sums W this means that for collections Ω = {ωk}, consisting of nonoverlapping intervals ωk ∪ [0,1) the following estimate is valid: where card denotes the number of elements, and c is an absolute positive constant. The exact value of the best absolute constant к in the estimate (which is due to G. H. Hardy and J. E. Littlewood) is discussed.

scholarly journals Solving Non-linear Polynomial Arithmetic via SAT Modulo Linear Arithmetic

2009 ◽  
pp. 294-305 ◽  
Author(s):  
Cristina Borralleras ◽  
Salvador Lucas ◽  
Rafael Navarro-Marset ◽  
Enric Rodríguez-Carbonell ◽  
Albert Rubio
Keyword(s):  
Linear Arithmetic ◽  
Linear Polynomial ◽  
Non Linear

scholarly journals Sommes friables d'exponentielles et applications

2015 ◽  
Vol 67 (3) ◽  
pp. 597-638 ◽  
Author(s):  
Sary Drappeau
Keyword(s):  
Asymptotic Formula ◽  
Saddle Point ◽  
Exponential Sums ◽  
Prime Factor ◽  
Character Sums ◽  
Contour Integration ◽  
Point Method ◽  
Saddle Point Method ◽  
Greatest Prime Factor

AbstractAn integer is said to be y–friable if its greatest prime factor is less than y. In this paper, we obtain estimates for exponential sums over y–friable numbers up to x which are non–trivial when y ≥ . As a consequence, we obtain an asymptotic formula for the number of y-friable solutions to the equation a + b = c which is valid unconditionally under the same assumption. We use a contour integration argument based on the saddle point method, as developped in the context of friable numbers by Hildebrand and Tenenbaum, and used by Lagarias, Soundararajan and Harper to study exponential and character sums over friable numbers.

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