Editing Constructible polygon (section)

===Connection to Pascal's triangle===

Since there are five known Fermat primes, we know of 31 numbers that are products of distinct Fermat primes, and hence we know of 31 constructible odd-sided regular polygons. These are 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, [[65537-gon|65537]], 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295 {{OEIS|id=A045544}}. As [[John Conway]] commented in ''The Book of Numbers'', these numbers, when written in [[binary number|binary]], are equal to the first 32 rows of the [[Modular arithmetic|modulo]]-2 [[Pascal's triangle]], minus the top row, which corresponds to a [[monogon]]. (Because of this, the ''1''s in such a list form an approximation to the [[Sierpiński triangle]].) This pattern breaks down after this, as the next Fermat number is composite (4294967297 = 641&thinsp;×&nbsp;6700417), so the following rows do not correspond to constructible polygons. It is unknown whether any more Fermat primes exist, and it is therefore unknown how many odd-sided constructible regular polygons exist. In general, if there are ''q'' Fermat primes, then there are 2<sup>''q''</sup>−1 {{nowrap|odd-sided}} regular constructible polygons.