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 20190127 11:57:32 