# Isqrt integer square root of X

Built-in:

```ruby
var n = 1234
say n.isqrt
say n.iroot(2)
```

Explicit implementation for the integer k-th root of n:

```ruby
func rootint(n, k=2) {
    return 0 if (n == 0)
    var (s, v) = (n, k - 1)
    loop {
        var u = ((v*s + (n // s**v)) // k)
        break if (u >= s)
        s = u
    }
    s
}
```

Implementation of integer square root of n (using the quadratic residue algorithm):

```ruby
func isqrt(x) { var (q, r) = (1, 0); while (q <= x) { q <<= 2 }
    while (q > 1) { q >>= 2; var t = x-r+q; r >>= 1
        if (t >= 0) { (x, r) = (t, r+q) } } r }

say isqrt.map(0..65).join(' '); printf("\n")

for n in (1..73 `by` 2) {
    printf("isqrt(7^%-2d): %42s\n", n, isqrt(7**n).commify) }
```

#### Output:

```
0 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8

isqrt(7^1 ):                                          2
isqrt(7^3 ):                                         18
isqrt(7^5 ):                                        129
isqrt(7^7 ):                                        907
isqrt(7^9 ):                                      6,352
isqrt(7^11):                                     44,467
isqrt(7^13):                                    311,269
isqrt(7^15):                                  2,178,889
isqrt(7^17):                                 15,252,229
isqrt(7^19):                                106,765,608
isqrt(7^21):                                747,359,260
isqrt(7^23):                              5,231,514,822
isqrt(7^25):                             36,620,603,758
isqrt(7^27):                            256,344,226,312
isqrt(7^29):                          1,794,409,584,184
isqrt(7^31):                         12,560,867,089,291
isqrt(7^33):                         87,926,069,625,040
isqrt(7^35):                        615,482,487,375,282
isqrt(7^37):                      4,308,377,411,626,977
isqrt(7^39):                     30,158,641,881,388,842
isqrt(7^41):                    211,110,493,169,721,897
isqrt(7^43):                  1,477,773,452,188,053,281
isqrt(7^45):                 10,344,414,165,316,372,973
isqrt(7^47):                 72,410,899,157,214,610,812
isqrt(7^49):                506,876,294,100,502,275,687
isqrt(7^51):              3,548,134,058,703,515,929,815
isqrt(7^53):             24,836,938,410,924,611,508,707
isqrt(7^55):            173,858,568,876,472,280,560,953
isqrt(7^57):          1,217,009,982,135,305,963,926,677
isqrt(7^59):          8,519,069,874,947,141,747,486,745
isqrt(7^61):         59,633,489,124,629,992,232,407,216
isqrt(7^63):        417,434,423,872,409,945,626,850,517
isqrt(7^65):      2,922,040,967,106,869,619,387,953,625
isqrt(7^67):     20,454,286,769,748,087,335,715,675,381
isqrt(7^69):    143,180,007,388,236,611,350,009,727,669
isqrt(7^71):  1,002,260,051,717,656,279,450,068,093,686
isqrt(7^73):  7,015,820,362,023,593,956,150,476,655,802
```


---

# Agent Instructions: Querying This Documentation

If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://trizen.gitbook.io/sidef-lang/programming_tasks/i/isqrt_integer_square_root_of_x.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.