# What is known about evil primes?

An evil number is a positive integer $$n$$ that has an even number of $$1$$s in its binary expansion. Many theorems exist about evil numbers, the most known ones are probably those that involve the Thue-Morse sequence.

However, I find no information about prime numbers having an even number of $$1$$s in their binary expansion. What is known about such numbers?

While it is obvious that the asymptotic density of evil numbers is $$1/2$$, is there an equivalent result/conjecture concerning evil primes?

Finally, is there anything known about the sum of the reciprocals of evil primes? (For evil numbers see here.)

• – Barry Cipra Mar 11 at 12:36
• Note the OP asked a similar question about $4$ hours later at Sum of reciprocal of evil/odious numbers. – John Omielan Mar 11 at 17:36
• On another note: how can evil be a positive? – usiro Mar 11 at 20:11
• @usiro I don't understand the question – Klangen May 12 at 11:23
• It is just odd how they called these numbers with this adjective, but coming back to to main question, there is an interesting pattern based on a difference between consecutive terms - I would start with that... – usiro May 12 at 21:04

There is a very similar Mathoverflow question, in which it was shown that the ratio of odd-bit primes against even-bit primes approaches $$1/2$$. In fact, the rigourous proof can be found in