The name comes from the fact that de Polignac erroneously conjectured that every odd number can be expressed in that way.

Erdös proved that there are infinite such numbers, for example all the numbers of the form 1260327937 + 2863311360⋅k.

The smallest composite number in this class is 905, while the first square is 40401.

Roger Crocker proved that there are infinite odd numbers not of the form  , with    prime and  0$">. The first numbers of this kind are 1, 3, 5, 6495105, 848629545, 1117175145, 2544265305,...

The smallest 3 × 3 magic square whose entries are de Polignac numbers is

The first de Polignac numbers are 1, 127, 149, 251, 331, 337, 373, 509, 599, 701, 757, 809, 877, 905, 907, 959, 977, 997 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

A graph displaying how many de Polignac numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.