Polygonal numbers in Mizar

Se ha publicado un artículo de razonamiento formalizado en Mizar titulado Polygonal numbers in Mizar.

Su autor es Adam Grabowski (de la Universidad de Białystok, Polonia).

Su resumen es

In the article the formal characterization of triangular numbers (famous from [14] and words “EYPHKA! num = ∆ + ∆ + ∆”) [16] is given. Our primary aim was to formalize one of the items (42) from Wiedijk’s Top 100 Mathematical Theorems list, namely that the sequence of sums of reciprocals of triangular numbers converges to 2 and this Mizar representation was written in 2007. As the Mizar language evolved and attributes with arguments were implemented, we decided to extend these lines and we characterized polygonal numbers.

We formalized centered polygonal numbers, the connection between triangular and square numbers, and also some equalities involving Mersenne primes and perfect numbers. We gave also explicit formula to obtain from the polygonal number its ordinal index. Also selected congruences modulo 10 were enumerated. Our work basically covers the Wikipedia item for triangular numbers and
the Online Encyclopedia of Integer Sequences (http://oeis.org/A000217).

An interesting related result [15] could be the proof of Lagrange’s four-square theorem or Fermat’s polygonal number theorem [30].

El artículo se publicará en la revista Formalized Mathematics.