De Polignac numbers
De Polignac numbers is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
Alphonse de Polignac, a French mathematician in the 1800s, conjectured that every positive odd integer could be formed from the sum of a power of 2 and a prime number.
He was subsequently proved incorrect.
The numbers that fail this condition are now known as de Polignac numbers.
Technically 1 is a de Polignac number, as there is no prime and power of 2 that sum to 1. De Polignac was aware but thought that 1 was a special case. However.
127 is also fails that condition, as there is no prime and power of 2 that sum to 127.
As it turns out, de Polignac numbers are not uncommon, in fact, there are an infinite number of them.
- Task
- Find and display the first fifty de Polignac numbers.
- Stretch
- Find and display the one thousandth de Polignac number.
- Find and display the ten thousandth de Polignac number.
- See also
Raku
use List::Divvy;
use Lingua::EN::Numbers;
constant @po2 = (1..∞).map: 2 ** *;
my @dePolignac = lazy 1, |(2..∞).hyper.map(* × 2 + 1).grep: -> $n { all @po2.&upto($n).map: { !is-prime $n - $_ } };
say "First fifty de Polignac numbers:\n" ~ @dePolignac[^50]».&comma».fmt("%5s").batch(10).join: "\n";
say "\nOne thousandth: " ~ comma @dePolignac[999];
say "\nTen thousandth: " ~ comma @dePolignac[9999];
- Output:
First fifty de Polignac numbers:
1 127 149 251 331 337 373 509 599 701
757 809 877 905 907 959 977 997 1,019 1,087
1,199 1,207 1,211 1,243 1,259 1,271 1,477 1,529 1,541 1,549
1,589 1,597 1,619 1,649 1,657 1,719 1,759 1,777 1,783 1,807
1,829 1,859 1,867 1,927 1,969 1,973 1,985 2,171 2,203 2,213
One thousandth: 31,941
Ten thousandth: 273,421