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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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poor explanation | Comment 2 of 4 |
first note (1 + √2)-100(√2 - 1)100=(1 - √2)100 since the power is even.

(1 + √2)100 is of the form a + b√2 

(1 - √2)100 is of the form a - b√2

So the sum is of the form 2a which is a whole number.

(5 + √5)100 is of the form c + d√5

Of the final product, we are only interested in the part after the decimal.  That is 2ac√5

Let the power be n (we want n=100) the values of a for n=0,1,2,... are the sequence 1,1,3,7,17,41,99.  (This is easy to show)  These numbers are the convergents to √5.  https://oeis.org/A001076

So it's not a surprise that a√5 is very close to a whole number.  These convergents are just over √5 for odd terms and just under for even terms.

Unfortunately, this explanation is lacking.  It doesn't show the effect of multiplying by 2c.  Why does this keep the result close to a whole number?

  Posted by Jer on 2019-01-27 11:57:32
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