In this paper, we study how to express all natural numbers when x, y, Z is taken as the natural value or integer value for an integer six tuple (a, B, C, D, e, f) polynomial x (AX + b) / 2 + y (CY + D) / 2 + Z (EZ + F) / 2.
For integers m = 3,4 The m-angle numbers introduced by the ancient Greek mathematicians according to the normal M-edge are given as follows: PM (n) = (m-2) n (n-1) / 2 + n (n = 0,1,2 ）In particular, the quaternion is the square number, and the trigonometric form is TN = n (n + 1) / 2 (n = 0,1,2 ）In 1638 Fermat claimed that every natural number (i.e. non negative integer) can be expressed as the sum of M angles. When m = 4, it was proved by Lagrange in 1770, when m = 3, it was solved by Gauss in 1796. When m ≥ 5, it was not proved by Cauchy until 1813. PM (x) also gives natural values when x is rounded. These are called generalized m-angle numbers.
In 1862, Liouville determined all seven positive integer triples (a, B, c) so that each natural number can be expressed in the form of ATX + BTY + CTZ, where x, y, Z belongs to the natural number set n = {0,1,2 }In 2005, the author initiated a study on how to mix square numbers with trigonometric numbers and how AX2 + BY2 + CTZ or AX2 + BTY + CTZ can represent all natural numbers. Through three papers by the author and his collaborators, this problem was finally solved in 2009.
In 2009, the author initiated a study on how three linear combination APIs (x) + bpj (y) + CPK (z) can represent all natural numbers (see SCI. China math. 58 (2015), 1367-1396), he proposed many conjectures and proved some results, for example, he guessed that each natural number can be expressed as a triangle number, a even quaternion (i.e. the sum of the square number) and a pentagonal number, and proved that this is correct when the pentagonal number is replaced by the generalized pentagonal number. South Korea's b-k. Oh proves part of the author's conjecture in this regard.
In this paper, we consider the sum of general quadratic polynomials of three single variables
x(ax+b)/2 + y(cy+d)/2 + z(ez+f)/2， （*）
Here a, B, C, D, e, f are integers, a ≥ C ≥ E & gt; 0, B & gt; - A, D & gt; - C, F & gt; - E, and a + B, C + D, e + F are all even numbers. If (*) can represent all natural numbers when x, y, Z are taken as natural numbers, we say that it (or the corresponding ordered six tuples (a, B, C, D, e, f)) is universal on n. Gauss's trigonometric theorem is equivalent to saying (1,1,1,1,1,1) is universal on n. In this paper, it is proved that (a, B, C, D, e, f) which is universal on N can only be found in 473 sextuples that we specifically listed. We also use various techniques to prove that 56 of them are indeed universal on N, and guess that the remaining candidates (see the appendix of this paper) are also universal on n.
In the case of 0 ≤ B & lt; a, 0 ≤ D & lt; C, 0 ≤ F & lt; E, if the neutralization formula of (*) can represent all natural numbers when the integral value of X, y, Z is taken, we say that it (or the corresponding ordered six tuples (a, B, C, D, e, f)) is universal on Z. In this paper, all the six tuple candidates (a, B, C, D, e, f) are determined, and some of them are proved to be universal in Z by using the theory of ternary quadratic form.
Let's use a specific example to illustrate. The author conjectures that (1,1,3,1,5,1) is universal on N, that is, polynomial
P(x,y,z) = x(x+1)/2 + y(3y+1)/2+ z(5z+1)/2
When x, y, Z take the natural number, it can represent all the natural numbers. We can't prove this and announce a reward of $135, but we prove that P (x, y, z) can represent any natural number when x, y, Z is an integral value.
The representation of natural numbers is the central topic of number theory. This paper develops a huge plan. It will be a long and arduous task to prove that the remaining ten thousand universal candidates (on N or Z) are indeed universal. Using the existing theory of ternary quadratic form is far from enough to accomplish this task. This paper is expected to stimulate new methods or theories to deal with such problems.
About the author:
Sun Zhiwei, born in October 1965. He is now a professor and doctoral supervisor in the Department of mathematics, Nanjing University, the director of mathematics and Applied Mathematics in the Department of mathematics, and the deputy director of the combination and graph theory Professional Committee of China Mathematics Association. His research direction is combination number theory and addition combination.
He has won many honors and awards, such as the first young teacher award of the Ministry of education, the National Science Foundation for Distinguished Young Scholars and the government special allowance of the State Council. He has many innovative achievements in the field of combination and number theory. So far, he has published more than 100 academic papers in the famous foreign mathematics journal trans. Amer. Math. SOC and other SCI magazines. He also put forward many original mathematical conjectures, which attracted the attention and research of some internationally famous mathematicians.
Article information: [click the link below or read the original] Sun z-w. universal sums of three horizontal polymers. SCI China math, 2020, 63: 501-520, https://doi.org/10.1007/s11425-017-9354-4