How to find out the error when taking a root from a number?

Question

Discrete Math

I have been trying to solve this problem for several days now:

There is a number 1 346 274 334 462 616 052 (for example), I take the square root of it and get 1 160 290 624.999882 . I need to take only 1 160 290 624 and I can make myself a hint, but a small one - up to 2 characters and it is better not to do it. Question: is there a way to find out all the numbers after the dot, without these same numbers? We took the square root of 1 346 274 334 462 616 052 , got 1 160 290 624.999882 , removed the numbers after the decimal point, got 1 160 290 624 . And now we need to get the number back. I will be glad to all the advice, and the train of thought)

spoiler

Пробовал использовать буквенные обозначения, но их всеравно не хватает

Answer 1

Well, generally speaking, it is impossible:

N                       sqrt(N)
1346274334462616042     1160290624.9998817
1346274334462616043     1160290624.9998817
1346274334462616044     1160290624.9998817
1346274334462616045     1160290624.9998817
1346274334462616046     1160290624.9998817
1346274334462616047     1160290624.9998817
1346274334462616048     1160290624.9998817
1346274334462616049     1160290624.9998817
1346274334462616050     1160290624.9998817
1346274334462616051     1160290624.9998817
1346274334462616052     1160290624.9998817
1346274334462616053     1160290624.9998817
1346274334462616054     1160290624.9998817
1346274334462616055     1160290624.9998817
1346274334462616056     1160290624.9998817
1346274334462616057     1160290624.9998817
1346274334462616058     1160290624.9998817
1346274334462616059     1160290624.9998817
1346274334462616060     1160290624.9998817
1346274334462616061     1160290624.9998817
1346274334462616062     1160290624.9998817

Furthermore:

1346274334462616052 - 1160290624^2 = 2320306676
sqrt(1346274334462616052 - 20000000) = 1160290624.991263148...

those. 2+ billion numbers have the same integer part of the square root, 20+ million have the same 2 decimal places. It is impossible to use roughly 6-7 bits of information (two decimal places) to select exactly one number out of 2 billion