The fun fact isn't just there for fun
Find the number of ordered pairs of positive integers with prime, such that is an integer.
Fun fact: is the th prime number.
The answer is 999.
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1 solution
First, an important correction. The given Fun Fact is wrong as 3581 is the 501-st prime number not 500-th.
a + b = p (prime) (a + b) divides (ab + 1) ==> p divides {a(p - a) + 1} = a*p - (a^2 - 1)
Therefore, p must divide (a^2 - 1) = (a - 1)*(a + 1) where 1 <= a <= (p - 1). p divides (a - 1) only when a = 1 and p divides (a + 1) only when a = (p - 1).
Thus, for each prime p the good ordered pairs would be = {(1, p-1); (p-1, 1)}. We know (p-1) = 1 for p = 2. Thus, there is only one ordered pair for 2 which is (1, 1). For all the other primes, there are 2 ordered pairs each.
So, the expected number of ordered pairs = 1 + (500 - 1)*2 = 999