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Digit Hunting (Posted on 2019-01-26) Difficulty: 3 of 5
Find the third digit after the decimal point of

(2 + √5)100((1 + √2)100 + (1 + √2)-100)

No Solution Yet Submitted by Danish Ahmed Khan    
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re(2): poor explanation Comment 4 of 4 |
(In reply to re: poor explanation by Brian Smith)

So you say that


[(2 + √5)^100 + (2 - √5)^100] is a very big integer (no roots). 

Implying that (2 + √5)^100 and (2 - √5)^100 are complementary: if the decimal part of (2 + √5)^100 is r then the decimal part of  
(2 - √5)^100 is 1-r

But  (2 - √5)^100 is 0,236067^100 and is going to have 61 zeros before the first significative digit. Much more zeros than digits has the expresion (1 + √2)^100 + (1 + √2)^-100 which is the integer  (2^51)-1. 

Then there are more than 20 zeros to begin the decimal part of 
[(2 - √5)^100] * [(1 + √2)^100 + (1 + √2)^-100] 

So that there are more than 20 nines to open the decimal part of 
[(2 + √5)^100] * [(1 + √2)^100 + (1 + √2)^-100] 

Good trick!

Edited on January 28, 2019, 4:30 pm
  Posted by armando on 2019-01-28 16:26:52

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