Polite Number - Trapezoidal Numbers

Trapezoidal Numbers

If a polite representation starts with 1, the number so represented is a triangular number

Otherwise, it is the difference of two triangular numbers:

In the latter case, it is called a trapezoidal number. That is, a trapezoidal number is a polite number that has a polite representation in which all terms are strictly greater than one. The only polite numbers that may be non-trapezoidal are the triangular numbers with only one nontrivial odd divisor, because for those numbers, according to the bijection described earlier, the odd divisor corresponds to the triangular representation and there can be no other polite representations. Thus, polite non-trapezoidal numbers must have the form of a power of two multiplied by a prime number. As Jones and Lord observe, there are exactly two types of triangular numbers with this form:

1. the even perfect numbers 2n − 1(2n − 1) formed by the product of a Mersenne prime 2n − 1 with half the nearest power of two, and
2. the products 2n − 1(2n + 1) of a Fermat prime 2n + 1 with half the nearest power of two.

For instance, the perfect number 28 = 23 − 1(23 − 1) and the number 136 = 24 − 1(24 + 1) are both polite triangular numbers that are not trapezoidal. It is believed that there are finitely many Fermat primes (only five of which — 3, 5, 17, 257, and 65,537 — have been discovered), but infinitely many Mersenne primes, in which case there are also infinitely many polite non-trapezoidal numbers.