Patterns /Encodings
Binary Repunits
These are the numbers one short of a power of two — 3, 7, 15, 31, 63, 1023 — whose binary form is nothing but ones.
Every binary repunit is 2^k − 1, so there is exactly one at each bit length and the count in range is tiny. This is one of the scarcest traits on the site.
They are the classic off-by-one boundaries in computing: 255, 65535 and 4294967295 are all repunits, and all three mark the edge of a common integer type.
When a repunit's exponent is prime, the number is a candidate Mersenne prime — 3, 7, 31 and 127 all qualify. That connection is why these particular numbers have been studied for centuries.
Highest-scoring examples
The best-scoring numbers of four digits or more that carry this pattern. Shorter numbers are excluded here because their length alone dominates any score ranking.
Smallest examples
The first numbers in the range that match this pattern.