Talk:Decagonal number

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Number of factors of powers of certain form of a number and s-gonal numbers with s even

Polygonal Number Formula is n(n(s - 2) - (s - 4))/2.

(n - 1 + 1) (n(s - 2)/2 - (s - 4)/2)
=> (n - 1 + 1) (n (s - 4 + 2)/2 - (s - 4)/2)
=> (n - 1 + 1) (n ((s - 4)/2 + 1) - (s - 4)/2)
(s - 4)/2 = t, if s is even, t is a natural number.
=> (n - 1 + 1) (n * (t + 1) - t)
p^(n - 1 + 1 - 1) * q^(n * (t + 1) - t - 1)
=> p^(n - 1) * q^(n * (t + 1) - (t + 1))
=> p^(n - 1) * q^((t + 1) (n - 1))
=> {p * q^(t + 1)}^(n - 1)
So, s-gonal number for s even is number of factors of (n - 1) power of p * q^(t + 1) where t = (s - 4)/2. This is the generalization of the property of the Decagonal numbers given here. Thanks! 152.58.177.247 (talk) 03:19, 14 March 2025 (UTC)[reply]

Polygonal Numbers with s = 2 * (n^2 + 1) has finitely many perfect squares and the first non-trivial perfect squares in this family

For numbers of form 2(n^2 + 1), the polygonal numbers with that number of sides has finitely many squares. Uptill n = 7 (100-gonal numbers), the only non-trivial squares are the 34-gonal square 14^2 = 196 and the 74-gonal square 51^2 = 2601. 152.58.177.247 (talk) 03:27, 14 March 2025 (UTC)[reply]