Integer as Sum of Polygonal Numbers/Lemma 1
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Theorem
Let $n, m \in \N_{>0}$ such that $m \ge 3$.
Let $n < 116 m$.
Then $n$ can be expressed as a sum of at most $m + 2$ polygonal numbers of order $m + 2$.
Proof
From Closed Form for Polygonal Numbers:
- $\map P {m + 2, k} = \dfrac m 2 \paren {k^2 - k} + k = m T_{k - 1} + k$
Where $T_{k - 1}$ are triangular numbers.
The first few $\paren {m + 2}$-gonal numbers less than $116 m$ are:
- $0, 1, m + 2, 3 m + 3, 6 m + 4, 10 m + 5, 15 m + 6, 21 m + 7, 28 m + 8, 36 m + 9, 45 m + 10, 55 m + 11, 66 m + 12, 78 m + 13, 91 m + 14, 105 m + 15$
We show the expression of the first few numbers explicitly.
\(\ds 1\) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds 2\) | \(=\) | \(\ds 1 + 1\) | ||||||||||||
\(\ds \) | \(:\) | \(\ds \) | ||||||||||||
\(\ds m + 1\) | \(=\) | \(\ds 1 + 1 + \dots + 1\) | There are $m + 1$ $1$'s | |||||||||||
\(\ds m + 2\) | \(=\) | \(\ds \paren {m + 2}\) | ||||||||||||
\(\ds \) | \(:\) | \(\ds \) | ||||||||||||
\(\ds 2 m + 3\) | \(=\) | \(\ds \paren {m + 2} + 1 + 1 + \dots + 1\) | There are $m + 1$ $1$'s | |||||||||||
\(\ds 2 m + 4\) | \(=\) | \(\ds \paren {m + 2} + \paren {m + 2}\) | ||||||||||||
\(\ds \) | \(:\) | \(\ds \) | ||||||||||||
\(\ds 3 m + 2\) | \(=\) | \(\ds \paren {m + 2} + \paren {m + 2} + 1 + 1 + \dots + 1\) | There are $m - 2$ $1$'s | |||||||||||
\(\ds 3 m + 3\) | \(=\) | \(\ds \paren {3 m + 3}\) | ||||||||||||
\(\ds \) | \(:\) | \(\ds \) | ||||||||||||
\(\ds 4 m + 4\) | \(=\) | \(\ds \paren {3 m + 3} + 1 + 1 + \dots + 1\) | There are $m + 1$ $1$'s | |||||||||||
\(\ds 4 m + 5\) | \(=\) | \(\ds \paren {3 m + 3} + \paren {m + 2}\) | ||||||||||||
\(\ds \) | \(:\) | \(\ds \) | ||||||||||||
\(\ds 5 m + 5\) | \(=\) | \(\ds \paren {3 m + 3} + \paren {m + 2} + 1 + 1 + \dots + 1\) | There are $m$ $1$'s | |||||||||||
\(\ds 5 m + 6\) | \(=\) | \(\ds \paren {m + 2} + \paren {m + 2} + \paren {m + 2} + \paren {m + 2} + 1 + 1 + \dots + 1\) | There are $m - 2$ $1$'s | |||||||||||
\(\ds 5 m + 7\) | \(=\) | \(\ds \paren {3 m + 3} + \paren {m + 2} + \paren {m + 2}\) | ||||||||||||
\(\ds \) | \(:\) | \(\ds \) | ||||||||||||
\(\ds 6 m + 6\) | \(=\) | \(\ds \paren {3 m + 3} + \paren {m + 2} + \paren {m + 2} + 1 + 1 + \dots + 1\) | There are $m - 1$ $1$'s | |||||||||||
\(\ds 6 m + 7\) | \(=\) | \(\ds \paren {6 m + 4} + 1 + 1 + 1\) | ||||||||||||
\(\ds \) | \(:\) | \(\ds \) | ||||||||||||
\(\ds 7 m + 5\) | \(=\) | \(\ds \paren {6 m + 4} + 1 + 1 + \dots + 1\) | There are $m + 1$ $1$'s | |||||||||||
\(\ds 7 m + 6\) | \(=\) | \(\ds \paren {6 m + 4} + \paren {m + 2}\) | ||||||||||||
\(\ds \) | \(:\) | \(\ds \) | ||||||||||||
\(\ds 8 m + 6\) | \(=\) | \(\ds \paren {6 m + 4} + \paren {m + 2} + 1 + 1 + \dots + 1\) | There are $m$ $1$'s | |||||||||||
\(\ds 8 m + 7\) | \(=\) | \(\ds \paren {3 m + 3} + \paren {3 m + 3} + \paren {m + 2} + 1 + 1 + \dots + 1\) | There are $m - 1$ $1$'s | |||||||||||
\(\ds 8 m + 8\) | \(=\) | \(\ds \paren {6 m + 4} + \paren {m + 2} + \paren {m + 2}\) | ||||||||||||
\(\ds \) | \(:\) | \(\ds \) | ||||||||||||
\(\ds 9 m + 7\) | \(=\) | \(\ds \paren {6 m + 4} + \paren {m + 2} + \paren {m + 2} + 1 + 1 + \dots + 1\) | There are $m - 1$ $1$'s | |||||||||||
\(\ds 9 m + 8\) | \(=\) | \(\ds \paren {6 m + 4} + \paren {3 m + 3} + 1\) | ||||||||||||
\(\ds \) | \(:\) | \(\ds \) | ||||||||||||
\(\ds 10 m + 8\) | \(=\) | \(\ds \paren {6 m + 4} + \paren {3 m + 3} + 1 + 1 + \dots + 1\) | There are $m$ $1$'s | |||||||||||
\(\ds 10 m + 9\) | \(=\) | \(\ds \paren {10 m + 5} + 1 + 1 + 1 + 1\) | ||||||||||||
\(\ds \) | \(:\) | \(\ds \) | ||||||||||||
\(\ds 11 m + 10\) | \(=\) | \(\ds \paren {10 m + 5} + 1 + 1 + \dots + 1\) | There are $m + 1$ $1$'s |
and so on.
$\blacksquare$