Perfect Square Numbers

Rule, exact count and rarity

A perfect square is the result of multiplying a nonnegative integer by itself. The atlas includes both 0 = 0² and 1 = 1². A candidate qualifies when its square root is an integer, so the inclusive upper endpoint is checked rather than silently omitted. The distance between successive squares grows as the root increases: (k + 1)² − k² = 2k + 1. Squares therefore become farther apart along the number line, even though the pattern receives one frequency weight for the complete range. A square may also be a cube, a Fibonacci number or a digit pattern. These overlaps affect its complete score; membership in the square sequence alone does not give every square the same ranking.

The exact count, 1,001, comes from evaluating all 1,000,001 integers from 0 through 1,000,000, including both endpoints. It is not a sample or an estimate. Each matching integer is counted once for this category, even if it contains multiple qualifying segments. Its frequency of 0.1001% describes the chance that a uniformly chosen integer passes this one test. It is not the global score percentile of every member, and overlapping pattern counts must not be added as if they represented disjoint sets.

The pattern supplies −log₂(1,001 / 1,000,001) bits before its within-family weight is applied. Other matched patterns, digit sum, distinct-digit count and short-length signals can change the complete score. Consequently, members of this category need not share a ranking. The highest-scoring representative is 1, with a rarity score of 1,504 and Top 0.0002%. This uses its complete score across all features, not just this pattern. Equal scores are displayed in ascending numerical order.

Example numbers

The highest-scoring match

The highest-scoring representative is 1, with a rarity score of 1,504 and Top 0.0002%. This uses its complete score across all features, not just this pattern. Equal scores are displayed in ascending numerical order.