# [seqfan] Variant of the Polygonal Numbers Theorem

Koscak Koseki koscak.koseki at gmail.com
Mon Feb 14 00:22:19 CET 2022

```Fermat polygonal number theorem states that *all* positive integers can be
written as a sum of at most k numbers of the form (k-2)*n*(n-1)/2+n, called
the k-gonal numbers, n a positive integer and k greater than or equal to 3.

But if one weakens the assumption to only require that all *sufficiently
large integers* are representable, then http://www.jstor.org/stable/2371077
Theorem 1, shows that if k=4j with j>=2 then one needs 5 such numbers, and
for all others at most 4 are required.

A118278 is the (conjectured) largest number that is not the sum of 3
k-gonal numbers, -1 if there is no such number, and zero if all positive
integers are the sum of at most 3 k-gonal numbers.

a(k) is the (conjectured, by brute force calculation upto .5*10^6) largest
number that is not the sum of 4 k-gonal numbers, -1 if there is no such
number, and zero if all positive integers are the sum of at most 4 k-gonal
numbers. a(k) for k=3,4,5,6,7,8,9,10,... is:
0,0,89,130,1003,-1,2894,2146,4765,-1,6906,11466,14057,-1,22296,24594,34913,-1,51301,55772,60551,-1,72335,65238,100206,-1,123097,117480,

a(k) is the (conjectured, by brute force calculation upto .5*10^6) largest
number that is not the sum of 5 k-gonal numbers. a(k) for
k=8,12,16,20,24,28,32,36,40,44,48,52,... (ie multiples of 4 greater than or
equal to 8) is:
188,821,1972,4788,9484,14064,20548,29952,50772,61528,77044,84833,102741,121368,

Koscak

```