Step by Step solution about How to resolve the algorithm Isqrt (integer square root) of X step by step in the Go programming language

. Overview of the Code: This Go program calculates the integer square roots of various numbers and presents the results in a formatted manner.

. Key Concepts:

• Big Integer (big.Int): The big package provides an arbitrary-precision integer type (big.Int) that can represent integers of unlimited size.
• Square Root: The square root of a number is another number that, when multiplied by itself, produces the original number.

. Main Function:

• main:
  • Executes the program.
  • Prints integer square roots of numbers from 0 to 65 and of powers of 7 up to 7^73.

. Functions:

• isqrt:
  • Computes the integer square root of a given non-negative big.Int number x.
  • Uses a binary search method to converge on the integer square root.
• commatize:
  • Formats a string by adding commas every three characters to improve readability of large numbers.

. Calculations and Output:

• The program calculates and prints the integer square roots of numbers from 0 to 65.
• It also calculates and prints the integer square roots of powers of 7 from 7^1 to 7^73. The output is formatted to align the power, the power of 7, and the integer square root.

. Example Output:

``` The integer square roots of integers from 0 to 65 are: 0 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 7 7 7 7 7 7 8 8 8 8 8 9 9 9 9 9 9 10 10 10 10 10 10 11 11 11 11 11 11 12 12 12 12 12 12 13 13 13 13 13 13 14 14 14 14 14 14 15 15 15

The integer square roots of powers of 7 from 7^1 up to 7^73 are:

power 3 7 ^ power integer square root

1 7 49 7 3 343 1,471 39 5 1,326 5,886 77 7 5,153 24,016 153 9 19,601 83,988 233 11 73,505 348,352 317 13 275,625 1,458,753 397 15 1,036,801 4,928,032 477 17 3,874,201 16,817,993 557 19 14,542,891 59,573,672 637 21 54,295,329 224,167,681 717 23 202,101,921 853,471,744 797 25 756,855,015 3,332,419,937 877 27 2,814,749,761 13,056,461,904 957 29 10,485,762,977 41,452,883,217 1037 31 39,100,263,169 161,061,273,696 1117 33 146,254,912,209 582,271,104,113 1197 35 547,561,516,225 2,181,736,125,744 1277 37 2,041,523,010,945 8,171,956,510,977 1357 39 7,599,796,076,673 29,303,275,625,744 1437 41 28,239,184,301,313 106,985,737,282,577 1517 43 104,956,737,205,217 396,540,551,341,088 1597 45 389,827,948,020,097 1,476,704,397,284,352 1677 47 1,442,310,602,481,025 5,453,665,191,754,249 1757 49 5,355,293,209,925,505 20,441,215,975,020,817 1837 51 19,986,577,839,730,225 75,930,399,680,077,248 1917 53 74,348,324,213,540,625 279,709,052,720,285,888 1997 55 275,573,195,600,962,505 1,034,210,550,801,062,912 2077 57 1,022,106,382,900,787,201 3,832,435,095,203,944,448 2157 59 3,797,069,744,100,634,081 14,406,812,382,815,777,856 2237 61 14,123,138,712,908,211,201 53,376,245,531,263,090,432 2317 63 52,474,411,385,016,632,321 197,529,806,245,052,473,856 2397 65 194,902,085,825,066,240,001 735,299,224,900,209,925,376 2477 67 724,184,347,250,264,426,241 2,716,760,533,600,799,749,632 2557 69 2,696,7

package main

import (
    "fmt"
    "log"
    "math/big"
)

var zero = big.NewInt(0)
var one = big.NewInt(1)

func isqrt(x *big.Int) *big.Int {
    if x.Cmp(zero) < 0 {
        log.Fatal("Argument cannot be negative.")
    }
    q := big.NewInt(1)
    for q.Cmp(x) <= 0 {
        q.Lsh(q, 2)
    }
    z := new(big.Int).Set(x)
    r := big.NewInt(0)
    for q.Cmp(one) > 0 {
        q.Rsh(q, 2)
        t := new(big.Int)
        t.Add(t, z)
        t.Sub(t, r)
        t.Sub(t, q)
        r.Rsh(r, 1)
        if t.Cmp(zero) >= 0 {
            z.Set(t)
            r.Add(r, q)
        }
    }
    return r
}

func commatize(s string) string {
    le := len(s)
    for i := le - 3; i >= 1; i -= 3 {
        s = s[0:i] + "," + s[i:]
    }
    return s
}

func main() {
    fmt.Println("The integer square roots of integers from 0 to 65 are:")
    for i := int64(0); i <= 65; i++ {
        fmt.Printf("%d ", isqrt(big.NewInt(i)))
    }
    fmt.Println()
    fmt.Println("\nThe integer square roots of powers of 7 from 7^1 up to 7^73 are:\n")
    fmt.Println("power                                    7 ^ power                                                 integer square root")
    fmt.Println("----- --------------------------------------------------------------------------------- -----------------------------------------")
    pow7 := big.NewInt(7)
    bi49 := big.NewInt(49)
    for i := 1; i <= 73; i += 2 {
        fmt.Printf("%2d %84s %41s\n", i, commatize(pow7.String()), commatize(isqrt(pow7).String()))
        pow7.Mul(pow7, bi49)
    }
}