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Extremely difficult question

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If x,y are positive real integers such that x can be expressed as an infinite sum of $$\frac{a}{\frac{b}{\frac{b}{\frac{b}{\cdots}+\frac{c}{\cdots}}+\frac{c}{\frac{b}{\cdots}+\frac{c}{\cdots}}}+\frac{b}{\frac{b}{\frac{b}{\cdots}+\frac{c}{\cdots}}+\frac{c}{\frac{b}{\cdots}+\frac{c}{\cdots}}}}$$ and a is the least prime number whicn has the property that any sum of two consecutive integers is divisible by it. C is defined recursively, where c=$$\frac{a}{\frac{b}{\frac{b}{\frac{b}{\cdots}+\frac{c}{\cdots}}+\frac{c}{\frac{b}{\cdots}+\frac{c}{\cdots}}}+\frac{b}{\frac{b}{\frac{b}{\cdots}+\frac{c}{\cdots}}+\frac{c}{\frac{b}{\cdots}+\frac{c}{\cdots}}}}$$. B has the property that it is the first prime number which is not a factor of any other prime number, and it also does not equal to C. Y has the property that the sum of an infinite geometric sequence with a scale factor of 1/2 with starting term one is equal to y plus $$e^{\pi i}$$. What is x+y?

Dec 19, 2018

Correction: If x,y are positive real integers such that x can be expressed as a continued fraction of $$\frac{a}{\frac{b}{\frac{b}{\frac{b}{\cdots}+\frac{x}{\cdots}}+\frac{x}{\frac{b}{\cdots}+\frac{x}{\cdots}}}+\frac{b}{\frac{b}{\frac{b}{\cdots}+\frac{x}{\cdots}}+\frac{x}{\frac{b}{\cdots}+\frac{x}{\cdots}}}}$$ and a is the least prime number which has the property that any sum of two nonnegative consecutive integers is divisible by it. B has the property that it is the first prime number which is not a factor of any other prime number, and it also does not equal to A. Y has the property that the sum of an infinite geometric sequence with a scale factor of 1/2 with starting term one is equal to y plus $$e^{\pi i}$$. What is x+y?