Automating John P. D'Angelo's method to study Complete Polynomial Sequences

By Shalosh B. Ekhad and Doron Zeilberger

.pdf   .tex

Written: Nov. 3, 2021.

In a recent intriguing article by complex geometer (and several complex variabler) John P. D'Angelo there was a surprising application of elementary (but very deep!) number theory to complex geometry. What D'Angelo needed was the largest integer not representable as sum of distinct values of triangular numbers (with analogous questions about other polynomial sequences). He demonstrated, in terms of a few lucid examples, how to determine these numbers, and then rigorously prove that they are indeed correct. In this short note we describe a Maple package that implements these ideas enabling, at least in principle, but often also in practice, to determine these numbers fully automatically. In fact, we show that the very same ideas can be turned into an algorithm that inputs an arbitrary integer-generating polynomial, and positive integers a and C, and outputs the largest integer not representable as a sum of distinct values of that polynomial with argument ≥a in at least C different ways. (The cases that D'Angelo needed were a=0 and a=1 and C=1).

## Input and Output files for JPDA.txt

• If you want to see an article with the largest integer NOT representable as a sum of distinct values of (j+1)*(j+2)/2, for j>=j0, for j0 from 0 to 6, each with a completely rigorous proof, following John D'Angelo's method described in his nice paper,

the input file yields the output file

• If you want to see an article, following D'Angelo's method in the above-mentioned paper, that tells you the largest integer NOT representable as a sum of distinct values of (j+1)*(j+2)/2, for j>=j0, for j0 from 0 to 5, in at least C different ways for C from 1 to 5,

the input file yields the output file

• If you want to see an article, following D'Angelo's method in the above-mentioned paper, that tells you the largest integer NOT representable as a sum of distinct values of (j+1)*(j+2)*(j+3)/6, for j>=j0, for j0 from 0 to 5

the input file yields the output file

• If you want to see an article, following D'Angelo's method in the above-mentioned paper, that tells you the largest integer NOT representable as a sum of distinct values of (j+1)*(j+2)*(j+3)*(j+4)/24, for j>=j0, for j0 from 0 to 5

the input file yields the output file

• If you want to see an article, following D'Angelo's method in the above-mentioned paper, that tells you the largest integer NOT representable as a sum of distinct values of (j+1)*(j+2)*(j+3)*(j+4)*(j+5)/120, for j>=0 (it happens to be 120838) and for j>=1 (it happens to be 291217)

the input file yields the output file

• If you want to see a more terse article, describing (without detailed proofs) the largest integer not representable in at least C different ways as distinct sums of the polynomial p(j)=(j+1)*(j+2)/2 with j>=j0, for C from 1 to 5 and j0 from 1 to 5,

the input file yields the output file

• If you want to see an example of D'Angelo's algorithm to produce representations

the input file yields the output file

• If you want to see an article, following D'Angelo's method in the above-mentioned paper, that tells you the largest integer NOT representable as a sum of distinct values of j2, for j>=j0, for j0 from 0 to 5, in at least C different ways for C from 1 to 5,

the input file yields the output file

• If you want to see an article, following D'Angelo's method in the above-mentioned paper, that tells you the largest integer NOT representable as a sum of distinct values of j3, for j>=j0, for j0 from 0 to 5, in at least C different ways for C from 1 to 5,

the input file yields the output file

• If you want to the first 11 terms of the sequence: The largest positive ineteger not representable as a sum of distinct values of (j+1)(j+2)/2 with j ≥ i, for i=0,1,2,...,

the input file yields the output file

• If you want to the first 11 terms of the sequence: The largest positive ineteger not representable as a sum of distinct values of (j+1)(j+2)(j+3)/6 with j ≥ i, for i=0,1,2,...,

the input file yields the output file

• If you want to the first 11 terms of the sequence: The largest positive ineteger not representable as a sum of distinct values of (j+1)(j+2)(j+3)(j+4)/24 with j ≥ i, for i=0,1,2,...,

the input file yields the output file

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