An inventory of arithmetic puzzles | Tangente

The fascination of figurate numbers (whose properties are reviewed in the boxes) has given rise to numerous arithmetic problems. Many involve finding integers that belong to two distinct classes of figurate numbers. Of all the questions we might ask, perhaps the most enticing is this: are there finitely or infinitely many solutions?

Let's compile a partial—and biased—catalogue of these problems. The wide variety of figurate numbers offers scope for countless others. Calling all enthusiasts!

Think in terms of recurrence! -----------------

The answer is yes. The first few solutions are 0, 1, 36, 1,225, 41,616, 1,413,721, 48,024,900, 1,631,432,881, 55,420,693,056, 1,882,672,131,025… These satisfy a recurrence: if (*a_n )_n *_{≥ 0} denotes this sequence, then a₀ = 0, a₁ = 1, and, for n ≥ 2, *a_n = 34 a_n*_–1 – *a_n–*₂ + 2.