On Double 3-term Arithmetic Progressions
نویسندگان
چکیده
In this note we are interested in the problem of whether or not every increasing sequence of positive integers x1x2x3 · · · with bounded gaps must contain a double 3-term arithmetic progression, i.e., three terms xi, x j, and xk such that i+k = 2 j and xi+xk = 2x j. We consider a few variations of the problem, discuss some related properties of double arithmetic progressions, and present several results obtained by using RamseyScript, a high-level scripting language.
منابع مشابه
On rainbow 4-term arithmetic progressions
{sl Let $[n]={1,dots, n}$ be colored in $k$ colors. A rainbow AP$(k)$ in $[n]$ is a $k$ term arithmetic progression whose elements have different colors. Conlon, Jungi'{c} and Radoiv{c}i'{c} cite{conlon} prove that there exists an equinumerous 4-coloring of $[4n]$ which is rainbow AP(4) free, when $n$ is even. Based on their construction, we show that such a coloring of $[4n]$...
متن کاملA 43 Integers 14 ( 2014 ) on Double 3 - Term Arithmetic Progressions
In this note we are interested in the problem of whether or not every increasing sequence of positive integers x1x2x3 · · · with bounded gaps must contain a double 3-term arithmetic progression, i.e., three terms xi, xj , and xk such that i+ k = 2j and xi+xk = 2xj . We consider a few variations of the problem, discuss some related properties of double arithmetic progressions, and present severa...
متن کاملArithmetic Progressions on Conics.
In this paper, we look at long arithmetic progressions on conics. By an arithmetic progression on a curve, we mean the existence of rational points on the curve whose x-coordinates are in arithmetic progression. We revisit arithmetic progressions on the unit circle, constructing 3-term progressions of points in the first quadrant containing an arbitrary rational point on the unit circle. We als...
متن کاملArithmetic Progressions of Three Squares
In this list there is an arithmetic progression: 1, 25, 49 (common difference 24). If we search further along, another arithmetic progression of squares is found: 289, 625, 961 (common difference 336). Yet another is 529, 1369, 2209 (common difference 840). How can these examples, and all others, be found? In Section 2 we will use plane geometry to describe the 3-term arithmetic progressions of...
متن کاملA 43 INTEGERS 12 ( 2012 ) ARITHMETIC PROGRESSIONS IN THE POLYGONAL NUMBERS Kenneth
In this paper, we investigate arithmetic progressions in the polygonal numbers with a fixed number of sides. We first show that four-term arithmetic progressions cannot exist. We then describe explicitly how to find all three-term arithmetic progressions. Finally, we show that not only are there infinitely many three-term arithmetic progressions, but that there are infinitely many three-term ar...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2014