Use binary search for the least positive integer whose square reaches the lower bound:
test(i,r):
    lo = 1; hi = i
    while lo < hi:
        mid = lo + (hi-lo)//2
        if r*mid*mid < i*i:
            lo = mid+1
        else:
            hi = mid
    return r*lo*lo < (i+1)*(i+1)
The search interval is valid because . Once the least witness is located, the final strict upper test decides membership, by part (a). No recursion or approximate square root is used.

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