You might find J H Conway's book The Sensual (quadratic) Form <https://www.maths.ed.ac.uk/~v1ranick/papers/conwaysens.pdf> of some interest for this problem. In the introduction, Conway says
I have been interested in quadratic forms for many years, but keep on discovering new and simple ways to understand them. The "topograph " of the First Lecture makes the entire theory of binary quadratic forms so easy that we no longer need to think or prove theorems about these forms-just look! I
On Tue, May 7, 2019 at 7:05 PM Keith F. Lynch <kfl@keithlynch.net> wrote:
Dan Asimov <dasimov@earthlink.net> wrote:
I made a table of the Loeschian numbers Lo(K,L) for K >= L. and then tried to *connect each one with the next largest one* with a line segment on paper.
But it was hard to see any pattern in how they connect (below). So the question boils down to:
Is there a concise formula for the "next largest number" in a set like Lo = {Lo(K,L) | K, L in Z} ??? I.e., f(0) = 1, f(1) = 3, f(3) = 4, ....
I wouldn't think so. There are lots of examples of two-dimensional integer sequences, aka tables, where there's a simple pattern to every row and to every column, but no pattern to all the numbers sorted in order.
Examples include:
* The multiplication table (excluding the 1s row and column). The entries are the composite numbers (A002808), which are of course exactly as patternless as the primes.
* Power numbers (A001597), i.e. the union of the squares, cubes, etc. Only in 2002 was it even proven that 8 and 9 are the only adjacent members of the set.
* Numbers in Pascal's triangle (other than the first two rows and columns), aka the binomial coefficients (A006987).
* Nontrivial polygonal or figurate numbers, i.e. the union of the triangular numbers (other than 1 and 3), the squares (other than 1 and 4), the pentagonal numbers (other than 1 and 5), etc. (A090466).
* Numbers that are a power of 2 plus a power of 3 (A004050). Similarly with any other two primes.
* Numbers that are a power of 2 times a power of 3 (A003586). Similarly with any other two primes.
Can anyone think of a nontrivial example of such a table sequence that *does* have a simple formula for finding the next one? I can't. (By "trivial," I think I just mean tables that contain all numbers or that contain only a small number of distinct numbers.)
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun