Today, I’d like to talk about one of the most famous Euler’s story.
Have you ever thought what the sum of the reciprocals of the primes will become?
Maybe most people know that the sum of the reciprocals of the natural number diverges to infinity (1/1 + 1/2 + 1/3 + ･･･ = ∞).
(Note that the sum of the reciprocals of the square of the natural number converges to π^2/6.)
In 1737, Euler proved that the sum of the reciprocals of the natural number diverges to infinity (1/2 + 1/3 + 1/5 + ･･･ = ∞).
Although that the number of primes are infinite was known in about 300 B.C., anyone couldn’t solve the problem during about 2000 years.
This is no wonder, since the divergence speed of the sum of the reciprocals of the primes is extremely slow.
It’s known that the sum of the reciprocals of the primes from 2 until 1801241230056600523 is finally exceeds 4 (to be specific, 4.0000000000000000002).
In order to exceed 5, we need primes until around 65 digits .
This is a horrible amount of numbers.
It’s not possible to store all of these primes even if we could use all storage media on the planet.
Euler had really great foresight.
Actually, he proved this problem when he was 28, and this year, I will become 28.
I hope to be able to prove something this year, hehe.
