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Joined 2026-09-21 Message
Incoming links: Bit length
Integer root bound Created 2026-10-20 Updated 2026-11-05
An integer root bound brackets an integer nth root using exact inequalities. If has bit length and , thenIndeed, ; taking positive real th roots gives the bounds. They supply an initial interval for binary search.
For and , the bit length is ten, giving . Testing nearby powers then finds the exact value . Using inequalities rather than rounded roots avoids excluding a perfect power at a search boundary.
Store a second array For the second loop, write . Immediately before processing , its loop invariant isInitially the failed squaring guard gives ; this also holds when , since then . The test succeeds exactly when . If successful, adding selects that binary digit; otherwise the upper bound tightens to . Either way the invariant holds for the next smaller . At , , so
weight with weight[k] = 2^k. Descending through the binary digits of the answer gives this greedy algorithm:PROCEDURE FastLog(b, n: INTEGER): INTEGER;
VAR b2, weight: ARRAY enough OF INTEGER;
top, k, i, p: INTEGER;
BEGIN
b2[0] := b; weight[0] := 1; top := 0;
WHILE b2[top]*b2[top] <= n DO
b2[top+1] := b2[top]*b2[top];
weight[top+1] := weight[top]+weight[top];
top := top+1
END;
i := 0; p := 1; k := top;
WHILE k >= 0 DO
IF p*b2[k] <= n THEN
p := p*b2[k]; i := i+weight[k]
END;
k := k-1
END;
RETURN i
END FastLog;i = L.There are squarings and descending tests. For fixed , this is arithmetic operations, hence logarithmic in the bit length of , and also within the weaker bound. This counts arithmetic operations on integers as unit cost; bit complexity would also charge for the growing operands. No division is used.