Substring search
Find where any digit-string first appears in √2 — and how many times it occurs in the first 2,000,000 digits, with the earliest positions listed.
Root2-Search
Search the first 2,000,000 decimal digits of √2 — the square root of two.
What √2 is
√2 is the positive number that, multiplied by itself, equals 2. Its decimal expansion begins 1.41421356… and never terminates or repeats: √2 is irrational — it cannot be written as a ratio of two whole numbers, a fact proved by the ancient Greeks. Because the digits go on forever without a pattern, the decimal expansion of √2 is a rich, fixed sequence to search.
Root2-Search indexes the fractional digits — the digits after the decimal point. Position 1 is the first fractional digit, 4; position 2 is 1; position 3 is 4; and so on.
What the tool does
Root2-Search runs entirely in your browser. The digits of √2 are downloaded once and cached; every query is then computed locally over the first 2,000,000 fractional digits. There is no backend — nothing you search is sent to a server.
Sum of a range
Sum the digits between two positions, from A to B. Returns both the regular sum and the squared sum (each digit²).
Sum-window search
Find every position where a sliding window of L consecutive digits sums to a target value — under the regular sum (n¹) or the squared sum (n²).
Composite queries
Require several sum and sequence conditions to hold at the same starting position at once. The scan runs on your CPU over the cached corpus.
Position analyzer
Click any returned position to see its prime factors, triangular / square / star indices, digital root, and sum-of-thousands.
Probability view
For a substring result, estimate how often a string of that length is expected to occur under a uniform-digit null model — and verify it with a built-in Monte-Carlo simulation.
Data source & provenance
The digits were computed by exact integer arithmetic: taking the integer square root of 2 · 102N — that is, isqrt(2 · 102N) — yields the first N fractional digits of √2 with no floating-point rounding. The result was then independently re-verified at higher precision and matched byte-for-byte.
You can reproduce the corpus from first principles with any arbitrary-precision library.
Reproduce the first N fractional digits (Python)
from math import isqrt
N = 2_000_000
digits = str(isqrt(2 * 10**(2*N)))[1:] # drop the leading integer '1'
assert digits[:8] == "41421356"
N = 2_000_000
digits = str(isqrt(2 * 10**(2*N)))[1:] # drop the leading integer '1'
assert digits[:8] == "41421356"
Verified, not estimated.
Every search runs over the actual digit array. Statistical estimation is offered only as an optional probability view — it is never substituted for a real scan of the corpus.
Questions
How many digits can I search?
The first 2,000,000 fractional digits of √2. All substring, sum, window, range, and composite queries operate over that corpus.
Does anything run on a server?
No. The digit file is served as a static asset and cached by your browser; every computation happens locally on your device. Sign-in is the only network call, and it is optional.
What does “position 1” mean?
Positions count the fractional digits, starting from the first digit after the decimal point. For √2 = 1.41421356…, position 1 is 4, position 2 is 1, position 3 is 4, and so on.
What is the squared sum (n²)?
It is the same window or range sum, but each digit is squared before being added. The regular sum (n¹) adds the digits as-is; the squared sum (n²) adds their squares.