Ponder This

July 2017 - Solution

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July 2017 Challenge

This solution is based on Oleg Vlasii's solution to the challenge.

First, note that almost all numbers of the form sin(a)*sin(b) are irrational, and there is a very small probability of seeing the digit '0' more than 20%. Actually, Andreas Stiller checked and found that the irrational product having the most number of zeros in the first one million digits is

sin(73°)*sin(75°) = 0.92371946158776085019620980936715820661197418131312... has only 101,008 zeros.

But there are a few such products that are rational numbers:

sin(15)*sin(75),
sin(18)*sin(54),
sin(30)*sin(30),
sin(45)*sin(45),
sin(60)*sin(60).

To check the 863,172th digit, you can use the GMP library or simply ask WolframAlpha:
"863173th digit of sin(18*pi/180)" ?

sin(15), sin(30), sin(45), sin(60), and sin(75) have an 863172th digit of "0"; only sin(18) and sin(54) have "9" as their 863172th digit.
So, one can choose PONDER = 18+360*k (or 180-18+360*k); THIS = 54+360*m (or 180-54+360*k); for example, PONDER = 902538, THIS = 7614.

Thanks to Motty Porat for pointing us to the relevant reference: When is the (co)sine of a rational angle equal to a rational number? by Jorg Jahnel