Theorem: Integer as Sum of Polygonal Numbers/Lemma 1

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.
{{begin-eqn}}
{{eqn | l = 1
      | r = 1
}}
{{eqn | l = 2
      | r = 1 + 1
}}
{{eqn | o = :
}}
{{eqn | l = m + 1
      | r = 1 + 1 + \dots + 1
      | c = There are $m + 1$ $1$'s
}}
{{eqn | l = m + 2
      | r = \paren {m + 2}
}}
{{eqn | o = :
}}
{{eqn | l = 2 m + 3
      | r = \paren {m + 2} + 1 + 1 + \dots + 1
      | c = There are $m + 1$ $1$'s
}}
{{eqn | l = 2 m + 4
      | r = \paren {m + 2} + \paren {m + 2}
}}
{{eqn | o = :
}}
{{eqn | l = 3 m + 2
      | r = \paren {m + 2} + \paren {m + 2} + 1 + 1 + \dots + 1
      | c = There are $m - 2$ $1$'s
}}
{{eqn | l = 3 m + 3
      | r = \paren {3 m + 3}
}}
{{eqn | o = :
}}
{{eqn | l = 4 m + 4
      | r = \paren {3 m + 3} + 1 + 1 + \dots + 1
      | c = There are $m + 1$ $1$'s
}}
{{eqn | l = 4 m + 5
      | r = \paren {3 m + 3} + \paren {m + 2}
}}
{{eqn | o = :
}}
{{eqn | l = 5 m + 5
      | r = \paren {3 m + 3} + \paren {m + 2} + 1 + 1 + \dots + 1
      | c = There are $m$ $1$'s
}}
{{eqn | l = 5 m + 6
      | r = \paren {m + 2} + \paren {m + 2} + \paren {m + 2} + \paren {m + 2} + 1 + 1 + \dots + 1
      | c = There are $m - 2$ $1$'s
}}
{{eqn | l = 5 m + 7
      | r = \paren {3 m + 3} + \paren {m + 2} + \paren {m + 2}
}}
{{eqn | o = :
}}
{{eqn | l = 6 m + 6
      | r = \paren {3 m + 3} + \paren {m + 2} + \paren {m + 2} + 1 + 1 + \dots + 1
      | c = There are $m - 1$ $1$'s
}}
{{eqn | l = 6 m + 7
      | r = \paren {6 m + 4} + 1 + 1 + 1
}}
{{eqn | o = :
}}
{{eqn | l = 7 m + 5
      | r = \paren {6 m + 4} + 1 + 1 + \dots + 1
      | c = There are $m + 1$ $1$'s
}}
{{eqn | l = 7 m + 6
      | r = \paren {6 m + 4} + \paren {m + 2}
}}
{{eqn | o = :
}}
{{eqn | l = 8 m + 6
      | r = \paren {6 m + 4} + \paren {m + 2} + 1 + 1 + \dots + 1
      | c = There are $m$ $1$'s
}}
{{eqn | l = 8 m + 7
      | r = \paren {3 m + 3} + \paren {3 m + 3} + \paren {m + 2} + 1 + 1 + \dots + 1
      | c = There are $m - 1$ $1$'s
}}
{{eqn | l = 8 m + 8
      | r = \paren {6 m + 4} + \paren {m + 2} + \paren {m + 2}
}}
{{eqn | o = :
}}
{{eqn | l = 9 m + 7
      | r = \paren {6 m + 4} + \paren {m + 2} + \paren {m + 2} + 1 + 1 + \dots + 1
      | c = There are $m - 1$ $1$'s
}}
{{eqn | l = 9 m + 8
      | r = \paren {6 m + 4} + \paren {3 m + 3} + 1
}}
{{eqn | o = :
}}
{{eqn | l = 10 m + 8
      | r = \paren {6 m + 4} + \paren {3 m + 3} + 1 + 1 + \dots + 1
      | c = There are $m$ $1$'s
}}
{{eqn | l = 10 m + 9
      | r = \paren {10 m + 5} + 1 + 1 + 1 + 1
}}
{{eqn | o = :
}}
{{eqn | l = 11 m + 10
      | r = \paren {10 m + 5} + 1 + 1 + \dots + 1
      | c = There are $m + 1$ $1$'s
}}
{{end-eqn}}
and so on.
{{qed}}
Category:Integer as Sum of Polygonal Numbers
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