 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Digit Hunting (Posted on 2019-01-26) 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 No Rating Comments: ( Back to comment list | You must be logged in to post comments.) 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 Please log in:

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