Consecutive Triplets not Sum of Pentagonal Numbers

Theorem

The following triplets of consecutive positive integers are such that none is the sum of $3$ pentagonal numbers:

$\tuple {19, 20, 21}$

$\tuple {88, 89, 90}$

$\tuple {99, 100, 101}$

$\tuple {111, 112, 113}$

Proof

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Sources

• Feb. 1994: Richard K. Guy: Every Number is Expressible as the Sum of How Many Polygonal Numbers? (Amer. Math. Monthly Vol. 101, no. 2: pp. 169 – 172)  www.jstor.org/stable/2324367
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $88$