digit_sum_search.py

"""
There are a surprising number of integers where the square root can be written
as the sum of the digits of the original minus 2. Even more if you're willing
to group digits together in order to make the sum work. This program finds all
299 such examples for n < 10e9. Ultimately this works because the square root
of a number tends to have roughly half as many digits as the original number,
so if you break a number in half, you'll already be in the ballpark of the
square root. For example, `sqrt(442,225) = 665 = (442 + 225) - 2`. The minus
two is arbitary - this works for other offsets as well - but offset 2 gets
you six matches for n <= 289 so it looks impressive right out of the gate.
"""
import math
import sys
from typing import Union, List, Tuple, Iterator

class DigitSumSearch:
    """
    Scans for instances where `sqrt(n) = sum(digits of n) - 2` such as:
    `sqrt(25) = 5 = (2 + 5) - 2`.
    
    It can also group digits as needed to make the sum work, such as: 
    `sqrt(121) = 11 = (12 + 1) - 2`.
    """
    def __init__(
        self,
        offset: int = 2,
        max_digit_group_length: int = 5,
        max_digit_cache_length: int = 5
    ):
        self.offset = int(offset)
        self.max_digit_group_length = int(max_digit_group_length)
        self.max_digit_cache_length = int(max_digit_cache_length)
        self.digit_cache = {}
    
    def check_sum_of_digits(self, digits: Union[int, str], target: int) -> List[int]:
        """
        Uses dynamic programing to find groups of digits which sum to a target value.
        """
        digits = str(digits)
        
        def recursive(digits, target):

            # successful base case: used all digits and found exact match
            if len(digits) == 0 and target == 0:
                return []

            # digit sum is already too big so we can stop searching.
            if target < 0:
                return None

            # check the cache
            if (digits, target) in self.digit_cache:
                return self.digit_cache[(digits, target)]

            # search for possible digit groupings that might work.
            for length in range(1, self.max_digit_group_length + 1):
                if len(digits) >= length:
                    n = int(digits[:length])
                    result = recursive(digits[length:], target - n)
                    if result is not None:
                        answer = [n] + result
                        if len(answer) <= self.max_digit_cache_length:
                            self.digit_cache[(digits, target)] = answer
                        return answer

            # No solution possible
            if len(digits) <= self.max_digit_cache_length:
                self.digit_cache[(digits, target)] = None
            return None

        return recursive(digits, target)
    
    def check_pattern(self, n):
        """
        Check if the square root of n is equal to the sum of its digits minus the offset.
        """
        root = math.isqrt(n)
        if root * root != n:
            return None
        return self.check_sum_of_digits(str(n), root + self.offset)

    def search(self, a: int, b: int = None) -> Iterator[Tuple[int, List[int]]]:
        """
        Searches from a to b (inclusive) for matching patterns
        and returns them as pairs such as `(121, [12, 1])`.
        """
        # if only a single argument is provided, scan up to that number.
        if b is None:
            a, b = 1, a

        # search the given range
        i = int(math.ceil(math.sqrt(a)))
        while (n := i ** 2) <= b:
            grouped_digits = self.check_pattern(n)
            if grouped_digits:
                yield n, grouped_digits
            i += 1

    def print_search(self, a: int, b: int = None):
        """
        Calls search(a, b) and prints each match with human readable formatting.
        """
        count = 0
        for n, digit_groups in self.search(a, b):
            root = math.isqrt(n)
            digit_group_sum = " + ".join(f"{digit_group:,}" for digit_group in digit_groups)
            print(f"sqrt({n:,}) = {root:,} = ({digit_group_sum}) - {self.offset}")
            count += 1
        print(f"\nNumber of matches found: {count}")

if __name__ == '__main__':
    DigitSumSearch().print_search(10_000_000_000)

output.txt

sqrt(4) = 2 = (4) - 2
sqrt(25) = 5 = (2 + 5) - 2
sqrt(64) = 8 = (6 + 4) - 2
sqrt(121) = 11 = (12 + 1) - 2
sqrt(196) = 14 = (1 + 9 + 6) - 2
sqrt(289) = 17 = (2 + 8 + 9) - 2
sqrt(1,024) = 32 = (10 + 24) - 2
sqrt(1,225) = 35 = (12 + 25) - 2
sqrt(1,936) = 44 = (1 + 9 + 36) - 2
sqrt(2,500) = 50 = (2 + 50 + 0) - 2
sqrt(4,225) = 65 = (42 + 25) - 2
sqrt(4,624) = 68 = (4 + 62 + 4) - 2
sqrt(6,889) = 83 = (68 + 8 + 9) - 2
sqrt(7,396) = 86 = (73 + 9 + 6) - 2
sqrt(8,464) = 92 = (84 + 6 + 4) - 2
sqrt(9,025) = 95 = (90 + 2 + 5) - 2
sqrt(9,604) = 98 = (96 + 0 + 4) - 2
sqrt(10,201) = 101 = (102 + 0 + 1) - 2
sqrt(12,100) = 110 = (12 + 100) - 2
sqrt(17,161) = 131 = (1 + 71 + 61) - 2
sqrt(17,956) = 134 = (1 + 79 + 56) - 2
sqrt(26,896) = 164 = (2 + 68 + 96) - 2
sqrt(27,889) = 167 = (2 + 78 + 89) - 2
sqrt(50,176) = 224 = (50 + 176) - 2
sqrt(110,224) = 332 = (110 + 224) - 2
sqrt(112,225) = 335 = (112 + 225) - 2
sqrt(118,336) = 344 = (1 + 1 + 8 + 336) - 2
sqrt(196,249) = 443 = (196 + 249) - 2
sqrt(204,304) = 452 = (20 + 430 + 4) - 2
sqrt(241,081) = 491 = (2 + 410 + 81) - 2
sqrt(250,000) = 500 = (2 + 500 + 0 + 0) - 2
sqrt(310,249) = 557 = (310 + 249) - 2
sqrt(426,409) = 653 = (4 + 2 + 640 + 9) - 2
sqrt(442,225) = 665 = (442 + 225) - 2
sqrt(446,224) = 668 = (44 + 622 + 4) - 2
sqrt(466,489) = 683 = (4 + 664 + 8 + 9) - 2
sqrt(574,564) = 758 = (5 + 745 + 6 + 4) - 2
sqrt(597,529) = 773 = (5 + 9 + 752 + 9) - 2
sqrt(602,176) = 776 = (602 + 176) - 2
sqrt(786,769) = 887 = (7 + 867 + 6 + 9) - 2
sqrt(792,100) = 890 = (792 + 100) - 2
sqrt(802,816) = 896 = (80 + 2 + 816) - 2
sqrt(808,201) = 899 = (80 + 820 + 1) - 2
sqrt(868,624) = 932 = (868 + 62 + 4) - 2
sqrt(879,844) = 938 = (87 + 9 + 844) - 2
sqrt(885,481) = 941 = (8 + 854 + 81) - 2
sqrt(902,500) = 950 = (902 + 50 + 0) - 2
sqrt(913,936) = 956 = (9 + 13 + 936) - 2
sqrt(966,289) = 983 = (966 + 2 + 8 + 9) - 2
sqrt(972,196) = 986 = (972 + 1 + 9 + 6) - 2
sqrt(978,121) = 989 = (978 + 12 + 1) - 2
sqrt(984,064) = 992 = (984 + 0 + 6 + 4) - 2
sqrt(990,025) = 995 = (990 + 0 + 2 + 5) - 2
sqrt(996,004) = 998 = (996 + 0 + 0 + 4) - 2
sqrt(1,002,001) = 1,001 = (1,002 + 0 + 0 + 1) - 2
sqrt(1,062,961) = 1,031 = (10 + 62 + 961) - 2
sqrt(1,229,881) = 1,109 = (1 + 229 + 881) - 2
sqrt(1,249,924) = 1,118 = (124 + 992 + 4) - 2
sqrt(1,495,729) = 1,223 = (1 + 495 + 729) - 2
sqrt(1,771,561) = 1,331 = (1 + 771 + 561) - 2
sqrt(1,779,556) = 1,334 = (1 + 779 + 556) - 2
sqrt(2,301,289) = 1,517 = (230 + 1,289) - 2
sqrt(2,768,896) = 1,664 = (2 + 768 + 896) - 2
sqrt(2,778,889) = 1,667 = (2 + 778 + 889) - 2
sqrt(3,171,961) = 1,781 = (3 + 1,719 + 61) - 2
sqrt(3,301,489) = 1,817 = (330 + 1,489) - 2
sqrt(4,182,025) = 2,045 = (4 + 18 + 2,025) - 2
sqrt(6,822,544) = 2,612 = (68 + 2 + 2,544) - 2
sqrt(7,268,416) = 2,696 = (7 + 2,684 + 1 + 6) - 2
sqrt(9,229,444) = 3,038 = (92 + 2,944 + 4) - 2
sqrt(10,316,944) = 3,212 = (1 + 0 + 3,169 + 44) - 2
sqrt(10,903,204) = 3,302 = (10 + 90 + 3,204) - 2
sqrt(11,102,224) = 3,332 = (1,110 + 2,224) - 2
sqrt(11,122,225) = 3,335 = (1,112 + 2,225) - 2
sqrt(12,823,561) = 3,581 = (12 + 8 + 2 + 3,561) - 2
sqrt(13,235,044) = 3,638 = (132 + 3,504 + 4) - 2
sqrt(13,586,596) = 3,686 = (1 + 3,586 + 5 + 96) - 2
sqrt(13,675,204) = 3,698 = (1 + 3,675 + 20 + 4) - 2
sqrt(13,697,401) = 3,701 = (1 + 3,697 + 4 + 0 + 1) - 2
sqrt(14,737,921) = 3,839 = (1 + 47 + 3,792 + 1) - 2
sqrt(15,389,929) = 3,923 = (15 + 3,899 + 2 + 9) - 2
sqrt(16,540,489) = 4,067 = (1 + 6 + 5 + 4,048 + 9) - 2
sqrt(18,421,264) = 4,292 = (18 + 4,212 + 64) - 2
sqrt(19,439,281) = 4,409 = (1 + 9 + 4,392 + 8 + 1) - 2
sqrt(20,784,481) = 4,559 = (2 + 0 + 78 + 4,481) - 2
sqrt(21,446,161) = 4,631 = (2 + 14 + 4,616 + 1) - 2
sqrt(22,174,681) = 4,709 = (2 + 21 + 7 + 4,681) - 2
sqrt(22,714,756) = 4,766 = (2 + 2 + 7 + 1 + 4,756) - 2
sqrt(23,348,224) = 4,832 = (2 + 3 + 3 + 4,822 + 4) - 2
sqrt(23,522,500) = 4,850 = (2,352 + 2,500) - 2
sqrt(24,492,601) = 4,949 = (24 + 4,926 + 0 + 1) - 2
sqrt(24,910,081) = 4,991 = (2 + 4,910 + 0 + 81) - 2
sqrt(25,000,000) = 5,000 = (2 + 5,000 + 0 + 0 + 0) - 2
sqrt(26,522,500) = 5,150 = (2,652 + 2,500) - 2
sqrt(27,520,516) = 5,246 = (27 + 5,205 + 16) - 2
sqrt(28,665,316) = 5,354 = (28 + 6 + 6 + 5,316) - 2
sqrt(29,604,481) = 5,441 = (2 + 960 + 4,481) - 2
sqrt(32,205,625) = 5,675 = (32 + 20 + 5,625) - 2
sqrt(36,857,041) = 6,071 = (368 + 5,704 + 1) - 2
sqrt(40,246,336) = 6,344 = (4 + 0 + 2 + 4 + 6,336) - 2
sqrt(41,396,356) = 6,434 = (41 + 39 + 6,356) - 2
sqrt(43,665,664) = 6,608 = (4 + 36 + 6,566 + 4) - 2
sqrt(44,422,225) = 6,665 = (4,442 + 2,225) - 2
sqrt(44,462,224) = 6,668 = (444 + 6,222 + 4) - 2
sqrt(44,662,489) = 6,683 = (44 + 6,624 + 8 + 9) - 2
sqrt(45,670,564) = 6,758 = (45 + 6,705 + 6 + 4) - 2
sqrt(46,730,896) = 6,836 = (4 + 6,730 + 8 + 96) - 2
sqrt(48,469,444) = 6,962 = (4 + 8 + 4 + 6,944 + 4) - 2
sqrt(52,867,441) = 7,271 = (528 + 6,744 + 1) - 2
sqrt(56,685,841) = 7,529 = (5 + 6,685 + 841) - 2
sqrt(57,775,201) = 7,601 = (5 + 77 + 7,520 + 1) - 2
sqrt(60,777,616) = 7,796 = (6 + 0 + 7,776 + 16) - 2
sqrt(66,080,641) = 8,129 = (6 + 60 + 8,064 + 1) - 2
sqrt(66,961,489) = 8,183 = (6,696 + 1,489) - 2
sqrt(67,354,849) = 8,207 = (6 + 7,354 + 849) - 2
sqrt(68,194,564) = 8,258 = (6 + 8,194 + 56 + 4) - 2
sqrt(68,244,121) = 8,261 = (6 + 8,244 + 12 + 1) - 2
sqrt(71,961,289) = 8,483 = (7,196 + 1,289) - 2
sqrt(73,085,401) = 8,549 = (7 + 3 + 0 + 8,540 + 1) - 2
sqrt(77,880,625) = 8,825 = (7 + 7 + 8,806 + 2 + 5) - 2
sqrt(78,889,924) = 8,882 = (7,888 + 992 + 4) - 2
sqrt(79,049,881) = 8,891 = (7,904 + 988 + 1) - 2
sqrt(80,281,600) = 8,960 = (802 + 8,160 + 0) - 2
sqrt(82,119,844) = 9,062 = (8,211 + 9 + 844) - 2
sqrt(82,828,201) = 9,101 = (8,282 + 820 + 1) - 2
sqrt(82,882,816) = 9,104 = (8,288 + 2 + 816) - 2
sqrt(83,759,104) = 9,152 = (8 + 37 + 5 + 9,104) - 2
sqrt(85,359,121) = 9,239 = (85 + 35 + 9,121) - 2
sqrt(86,192,656) = 9,284 = (8 + 6 + 1 + 9,265 + 6) - 2
sqrt(87,086,224) = 9,332 = (8,708 + 622 + 4) - 2
sqrt(87,928,129) = 9,377 = (87 + 9,281 + 2 + 9) - 2
sqrt(90,935,296) = 9,536 = (90 + 9,352 + 96) - 2
sqrt(91,164,304) = 9,548 = (9,116 + 430 + 4) - 2
sqrt(91,795,561) = 9,581 = (9 + 17 + 9,556 + 1) - 2
sqrt(96,687,889) = 9,833 = (9,668 + 78 + 89) - 2
sqrt(96,746,896) = 9,836 = (9,674 + 68 + 96) - 2
sqrt(96,983,104) = 9,848 = (9 + 6 + 9,831 + 0 + 4) - 2
sqrt(97,337,956) = 9,866 = (9,733 + 79 + 56) - 2
sqrt(97,397,161) = 9,869 = (9,739 + 71 + 61) - 2
sqrt(98,049,604) = 9,902 = (9,804 + 96 + 0 + 4) - 2
sqrt(98,109,025) = 9,905 = (9,810 + 90 + 2 + 5) - 2
sqrt(98,168,464) = 9,908 = (9,816 + 84 + 6 + 4) - 2
sqrt(98,287,396) = 9,914 = (9,828 + 73 + 9 + 6) - 2
sqrt(98,346,889) = 9,917 = (9,834 + 68 + 8 + 9) - 2
sqrt(98,644,624) = 9,932 = (9,864 + 4 + 62 + 4) - 2
sqrt(98,704,225) = 9,935 = (9,870 + 42 + 25) - 2
sqrt(99,002,500) = 9,950 = (9,900 + 2 + 50 + 0) - 2
sqrt(99,121,936) = 9,956 = (9,912 + 1 + 9 + 36) - 2
sqrt(99,301,225) = 9,965 = (9,930 + 12 + 25) - 2
sqrt(99,361,024) = 9,968 = (9,936 + 10 + 24) - 2
sqrt(99,660,289) = 9,983 = (9,966 + 0 + 2 + 8 + 9) - 2
sqrt(99,720,196) = 9,986 = (9,972 + 0 + 1 + 9 + 6) - 2
sqrt(99,780,121) = 9,989 = (9,978 + 0 + 12 + 1) - 2
sqrt(99,840,064) = 9,992 = (9,984 + 0 + 0 + 6 + 4) - 2
sqrt(99,900,025) = 9,995 = (9,990 + 0 + 0 + 2 + 5) - 2
sqrt(99,960,004) = 9,998 = (9,996 + 0 + 0 + 0 + 4) - 2
sqrt(100,020,001) = 10,001 = (10,002 + 0 + 0 + 0 + 1) - 2
sqrt(102,010,000) = 10,100 = (102 + 0 + 10,000) - 2
sqrt(111,049,444) = 10,538 = (1 + 1 + 10,494 + 44) - 2
sqrt(132,618,256) = 11,516 = (1 + 3,261 + 8,256) - 2
sqrt(139,617,856) = 11,816 = (1 + 3,961 + 7,856) - 2
sqrt(177,715,561) = 13,331 = (1 + 7,771 + 5,561) - 2
sqrt(177,795,556) = 13,334 = (1 + 7,779 + 5,556) - 2
sqrt(201,412,864) = 14,192 = (2 + 0 + 14,128 + 64) - 2
sqrt(214,212,496) = 14,636 = (2,142 + 12,496) - 2
sqrt(246,615,616) = 15,704 = (24 + 66 + 15,616) - 2
sqrt(252,015,625) = 15,875 = (252 + 0 + 15,625) - 2
sqrt(277,688,896) = 16,664 = (2 + 7,768 + 8,896) - 2
sqrt(277,788,889) = 16,667 = (2 + 7,778 + 8,889) - 2
sqrt(349,615,204) = 18,698 = (3,496 + 15,204) - 2
sqrt(452,115,169) = 21,263 = (45 + 21,151 + 69) - 2
sqrt(682,254,400) = 26,120 = (682 + 25,440 + 0) - 2
sqrt(932,630,521) = 30,539 = (9 + 3 + 2 + 6 + 30,521) - 2
sqrt(993,132,196) = 31,514 = (99 + 31,321 + 96) - 2
sqrt(1,111,022,224) = 33,332 = (11,110 + 22,224) - 2
sqrt(1,111,222,225) = 33,335 = (11,112 + 22,225) - 2
sqrt(1,229,834,761) = 35,069 = (12 + 298 + 34,761) - 2
sqrt(1,235,100,736) = 35,144 = (1 + 2 + 35,100 + 7 + 36) - 2
sqrt(1,261,599,361) = 35,519 = (1 + 26,159 + 9,361) - 2
sqrt(1,305,015,625) = 36,125 = (1 + 30,501 + 5,625) - 2
sqrt(1,327,436,356) = 36,434 = (1 + 3 + 2 + 74 + 36,356) - 2
sqrt(1,359,101,956) = 36,866 = (1 + 35,910 + 1 + 956) - 2
sqrt(1,366,633,024) = 36,968 = (1 + 36,663 + 302 + 4) - 2
sqrt(1,369,296,016) = 37,004 = (1 + 36,929 + 60 + 16) - 2
sqrt(1,369,518,049) = 37,007 = (1 + 36,951 + 8 + 0 + 49) - 2
sqrt(1,369,962,169) = 37,013 = (1 + 36,996 + 2 + 1 + 6 + 9) - 2
sqrt(1,437,850,561) = 37,919 = (1 + 4 + 37,850 + 5 + 61) - 2
sqrt(1,441,037,521) = 37,961 = (1 + 441 + 0 + 37,521) - 2
sqrt(1,696,039,489) = 41,183 = (1,696 + 0 + 39,489) - 2
sqrt(1,786,414,756) = 42,266 = (1 + 786 + 41,475 + 6) - 2
sqrt(1,909,427,809) = 43,697 = (1 + 909 + 42,780 + 9) - 2
sqrt(1,932,042,025) = 43,955 = (1,932 + 0 + 42,025) - 2
sqrt(2,045,210,176) = 45,224 = (2 + 0 + 45,210 + 1 + 7 + 6) - 2
sqrt(2,239,844,929) = 47,327 = (2 + 2,398 + 44,929) - 2
sqrt(2,301,024,961) = 47,969 = (23,010 + 24,961) - 2
sqrt(2,373,248,656) = 48,716 = (23 + 7 + 32 + 48,656) - 2
sqrt(2,491,008,100) = 49,910 = (2 + 49,100 + 810 + 0) - 2
sqrt(2,499,100,081) = 49,991 = (2 + 49,910 + 0 + 0 + 81) - 2
sqrt(2,500,000,000) = 50,000 = (2 + 50,000 + 0 + 0 + 0 + 0) - 2
sqrt(2,555,504,704) = 50,552 = (25 + 55 + 50,470 + 4) - 2
sqrt(2,595,087,364) = 50,942 = (2 + 59 + 50,873 + 6 + 4) - 2
sqrt(2,630,048,656) = 51,284 = (2,630 + 0 + 48,656) - 2
sqrt(2,707,224,961) = 52,031 = (27,072 + 24,961) - 2
sqrt(2,752,366,369) = 52,463 = (27 + 52,366 + 3 + 69) - 2
sqrt(2,774,444,929) = 52,673 = (2 + 7,744 + 44,929) - 2
sqrt(2,866,852,849) = 53,543 = (28 + 668 + 52,849) - 2
sqrt(2,922,051,136) = 54,056 = (2,922 + 0 + 51,136) - 2
sqrt(3,054,551,824) = 55,268 = (30 + 54 + 55,182 + 4) - 2
sqrt(3,065,504,689) = 55,367 = (306 + 55,046 + 8 + 9) - 2
sqrt(3,357,854,809) = 57,947 = (3 + 3 + 57,854 + 80 + 9) - 2
sqrt(3,458,733,721) = 58,811 = (3 + 4 + 58,733 + 72 + 1) - 2
sqrt(3,553,590,544) = 59,612 = (3 + 553 + 59,054 + 4) - 2
sqrt(3,680,606,224) = 60,668 = (36 + 8 + 0 + 60,622 + 4) - 2
sqrt(3,686,068,369) = 60,713 = (3 + 6 + 8 + 60,683 + 6 + 9) - 2
sqrt(3,737,610,496) = 61,136 = (3 + 73 + 7 + 61,049 + 6) - 2
sqrt(4,167,864,481) = 64,559 = (4 + 1 + 67 + 8 + 64,481) - 2
sqrt(4,426,640,089) = 66,533 = (4 + 42 + 66,400 + 89) - 2
sqrt(4,444,222,225) = 66,665 = (44,442 + 22,225) - 2
sqrt(4,444,622,224) = 66,668 = (4,444 + 62,222 + 4) - 2
sqrt(4,446,622,489) = 66,683 = (444 + 66,224 + 8 + 9) - 2
sqrt(4,456,630,564) = 66,758 = (445 + 66,305 + 6 + 4) - 2
sqrt(4,510,062,649) = 67,157 = (4,510 + 0 + 62,649) - 2
sqrt(4,683,349,225) = 68,435 = (4 + 68,334 + 92 + 2 + 5) - 2
sqrt(4,683,759,844) = 68,438 = (4 + 68,375 + 9 + 8 + 44) - 2
sqrt(4,736,880,625) = 68,825 = (4 + 7 + 3 + 68,806 + 2 + 5) - 2
sqrt(4,806,926,224) = 69,332 = (48 + 0 + 69,262 + 24) - 2
sqrt(4,833,169,441) = 69,521 = (48 + 3 + 31 + 69,441) - 2
sqrt(4,890,065,041) = 69,929 = (4,890 + 0 + 65,041) - 2
sqrt(4,931,269,729) = 70,223 = (493 + 1 + 2 + 69,729) - 2
sqrt(4,973,070,400) = 70,520 = (49 + 73 + 0 + 70,400) - 2
sqrt(4,991,705,104) = 70,652 = (49 + 91 + 70,510 + 4) - 2
sqrt(5,048,670,916) = 71,054 = (50 + 4 + 86 + 70,916) - 2
sqrt(5,293,726,564) = 72,758 = (5 + 2 + 93 + 72,656 + 4) - 2
sqrt(5,367,320,644) = 73,262 = (5 + 3 + 6 + 73,206 + 44) - 2
sqrt(5,714,755,216) = 75,596 = (57 + 14 + 75,521 + 6) - 2
sqrt(5,734,275,625) = 75,725 = (57 + 3 + 42 + 75,625) - 2
sqrt(6,545,780,836) = 80,906 = (6 + 5 + 4 + 57 + 80,836) - 2
sqrt(6,610,015,204) = 81,302 = (66,100 + 15,204) - 2
sqrt(6,714,819,136) = 81,944 = (6 + 7 + 14 + 81,913 + 6) - 2
sqrt(6,779,381,569) = 82,337 = (677 + 93 + 81,569) - 2
sqrt(6,784,816,900) = 82,370 = (678 + 4 + 81,690 + 0) - 2
sqrt(6,791,243,281) = 82,409 = (6 + 79,124 + 3,281) - 2
sqrt(6,825,899,161) = 82,619 = (6 + 82,589 + 9 + 16 + 1) - 2
sqrt(6,983,443,489) = 83,567 = (69 + 83,443 + 48 + 9) - 2
sqrt(7,287,012,496) = 85,364 = (72,870 + 12,496) - 2
sqrt(7,328,558,449) = 85,607 = (7 + 3 + 2 + 85,584 + 4 + 9) - 2
sqrt(7,604,886,436) = 87,206 = (760 + 4 + 8 + 86,436) - 2
sqrt(7,898,587,876) = 88,874 = (7 + 8 + 985 + 87,876) - 2
sqrt(8,129,005,921) = 90,161 = (81 + 2 + 90,059 + 21) - 2
sqrt(8,248,998,976) = 90,824 = (824 + 89,989 + 7 + 6) - 2
sqrt(8,288,281,600) = 91,040 = (82,882 + 8,160 + 0) - 2
sqrt(8,539,238,464) = 92,408 = (8 + 5 + 3 + 92,384 + 6 + 4) - 2
sqrt(8,598,667,441) = 92,729 = (85,986 + 6,744 + 1) - 2
sqrt(8,710,862,224) = 93,332 = (87,108 + 6,222 + 4) - 2
sqrt(8,822,657,041) = 93,929 = (88,226 + 5,704 + 1) - 2
sqrt(8,894,941,969) = 94,313 = (8 + 8 + 94 + 94,196 + 9) - 2
sqrt(8,994,815,281) = 94,841 = (8 + 9 + 94,815 + 2 + 8 + 1) - 2
sqrt(9,070,086,169) = 95,237 = (9,070 + 0 + 86,169) - 2
sqrt(9,204,867,364) = 95,942 = (9,204 + 86,736 + 4) - 2
sqrt(9,285,635,044) = 96,362 = (92,856 + 3,504 + 4) - 2
sqrt(9,322,095,601) = 96,551 = (932 + 20 + 95,601) - 2
sqrt(9,401,629,444) = 96,962 = (94,016 + 2,944 + 4) - 2
sqrt(9,484,422,544) = 97,388 = (94,844 + 2 + 2,544) - 2
sqrt(9,794,862,961) = 98,969 = (97,948 + 62 + 961) - 2
sqrt(9,809,713,936) = 99,044 = (98,097 + 13 + 936) - 2
sqrt(9,812,685,481) = 99,059 = (98,126 + 854 + 81) - 2
sqrt(9,823,386,769) = 99,113 = (98,233 + 867 + 6 + 9) - 2
sqrt(9,845,997,529) = 99,227 = (98,459 + 9 + 752 + 9) - 2
sqrt(9,848,974,564) = 99,242 = (98,489 + 745 + 6 + 4) - 2
sqrt(9,863,866,489) = 99,317 = (98,638 + 664 + 8 + 9) - 2
sqrt(9,869,826,409) = 99,347 = (98,698 + 2 + 640 + 9) - 2
sqrt(9,900,250,000) = 99,500 = (99,002 + 500 + 0 + 0) - 2
sqrt(9,902,041,081) = 99,509 = (99,020 + 410 + 81) - 2
sqrt(9,913,985,761) = 99,569 = (991 + 3 + 98,576 + 1) - 2
sqrt(9,931,318,336) = 99,656 = (99,313 + 1 + 8 + 336) - 2
sqrt(9,955,250,176) = 99,776 = (99,552 + 50 + 176) - 2
sqrt(9,966,627,889) = 99,833 = (99,666 + 2 + 78 + 89) - 2
sqrt(9,967,226,896) = 99,836 = (99,672 + 2 + 68 + 96) - 2
sqrt(9,973,217,956) = 99,866 = (99,732 + 1 + 79 + 56) - 2
sqrt(9,973,817,161) = 99,869 = (99,738 + 1 + 71 + 61) - 2
sqrt(9,978,012,100) = 99,890 = (99,780 + 12 + 100) - 2
sqrt(9,979,810,201) = 99,899 = (99,798 + 102 + 0 + 1) - 2
sqrt(9,980,409,604) = 99,902 = (99,804 + 0 + 96 + 0 + 4) - 2
sqrt(9,981,009,025) = 99,905 = (99,810 + 0 + 90 + 2 + 5) - 2
sqrt(9,981,608,464) = 99,908 = (99,816 + 0 + 84 + 6 + 4) - 2
sqrt(9,982,807,396) = 99,914 = (99,828 + 0 + 73 + 9 + 6) - 2
sqrt(9,983,406,889) = 99,917 = (99,834 + 0 + 68 + 8 + 9) - 2
sqrt(9,986,404,624) = 99,932 = (99,864 + 0 + 4 + 62 + 4) - 2
sqrt(9,987,004,225) = 99,935 = (99,870 + 0 + 42 + 25) - 2
sqrt(9,990,002,500) = 99,950 = (99,900 + 0 + 2 + 50 + 0) - 2
sqrt(9,991,201,936) = 99,956 = (99,912 + 0 + 1 + 9 + 36) - 2
sqrt(9,993,001,225) = 99,965 = (99,930 + 0 + 12 + 25) - 2
sqrt(9,993,601,024) = 99,968 = (99,936 + 0 + 10 + 24) - 2
sqrt(9,996,600,289) = 99,983 = (99,966 + 0 + 0 + 2 + 8 + 9) - 2
sqrt(9,997,200,196) = 99,986 = (99,972 + 0 + 0 + 1 + 9 + 6) - 2
sqrt(9,997,800,121) = 99,989 = (99,978 + 0 + 0 + 12 + 1) - 2
sqrt(9,998,400,064) = 99,992 = (99,984 + 0 + 0 + 0 + 6 + 4) - 2
sqrt(9,999,000,025) = 99,995 = (99,990 + 0 + 0 + 0 + 2 + 5) - 2
sqrt(9,999,600,004) = 99,998 = (99,996 + 0 + 0 + 0 + 0 + 4) - 2

Number of matches found: 299