2
$\begingroup$

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.)

$\endgroup$
  • 1
    $\begingroup$ See mathoverflow.net/questions/44561/odd-bit-primes-ratio $\endgroup$ – Barry Cipra Mar 11 at 12:36
  • 1
    $\begingroup$ Note the OP asked a similar question about $4$ hours later at Sum of reciprocal of evil/odious numbers. $\endgroup$ – John Omielan Mar 11 at 17:36
  • 1
    $\begingroup$ On another note: how can evil be a positive? $\endgroup$ – usiro Mar 11 at 20:11
  • $\begingroup$ @usiro I don't understand the question $\endgroup$ – Klangen May 12 at 11:23
  • $\begingroup$ 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... $\endgroup$ – usiro May 12 at 21:04
0
$\begingroup$

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

C. Mauduit and J. Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers, Ann. Math.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.