Race between primes of the forms 4k and 1 and 4k and 3
7 days ago
- #prime numbers
- #number theory
- #Chebyshev bias
- An odd prime p can be expressed as a sum of two squares if and only if p is of the form 4k + 1.
- Mersenne primes, which are easier to find due to the Lucas-Lehmer test, are typically of the form 4k + 3.
- The ratio of primes congruent to 3 mod 4 versus 1 mod 4 converges to 1 as n approaches infinity.
- The difference between the counts of primes congruent to 3 mod 4 and 1 mod 4, known as the lead function, oscillates infinitely but is often positive.
- The lead function changes sign at specific primes, listed in OEIS sequence A038691.