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.
How long might binary repunits take?
1 in 55,555.6 per roll · 55,556 rolls on average.
At one daily specimen a day, the mean is about 152.1 years. A 50% chance takes 38,508 rolls; 90% takes 127,921.
7 daily rolls
1 in 7,937
30 daily rolls
1 in 1,852
365 daily rolls
1 in 153
1 in 1,852 chance of at least one match in 30 independent rolls.
Waiting times are averages, not deadlines. Previous misses do not increase the next roll’s chance. Only Daily Specimens stamp your Atlas.
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.