The Mystery of Primes

Number Theory Level pending

let the series \(a_1,a_2,a_3,a_4 ...\) , be the series of prime numbers starting with 3.

What will be that value of the infinite series (1 ■ \(\frac{1}{a_1}\))(1 ■ \(\frac{1}{a_2}\))(1 ■ \(\frac{1}{a_3}\))... given these rules

You must fill the black square with the appropriate sign according to these rules:

1) if the value of \(a_n\) can be represented by 4k+1 where k is a positive integer, replace the black box with a PLUS sign

2) if not then replace the black box with a MINUS sign

Shown below are the first 3 terms of the infinite series (1-\(\frac{1}{3}\))(1+\(\frac{1}{5}\))(1-\(\frac{1}{7}\))

The value of the infinite series (1 ■ \(\frac{1}{a_1}\))(1 ■ \(\frac{1}{a_2}\))(1 ■ \(\frac{1}{a_3}\))... can be represented as \(\frac{x}{y}\) where x and y are real numbers and \(\frac{x}{y}\) is in lowest terms. What is the value of [x+y]?

[x] represents the greatest integer function.

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