Chebyshev's bias

In number theory, Chebyshev's bias is the phenomenon that most of the time, there are more primes of the form 4k + 3 than of the form 4k + 1, up to the same limit.

This phenomenon was first observed by Russian mathematician Pafnuty Chebyshev in 1853.

Let π(x; n, m) denote the number of primes of the form nk + m up to x.

By the prime number theorem (extended to arithmetic progression), That is, half of the primes are of the form 4k + 1, and half of the form 4k + 3.

This, however, is not supported by numerical evidence — in fact, π(x; 4, 3) > π(x; 4, 1) occurs much more frequently.

In general, if 0 < a, b < n are integers, gcd(a, n) = gcd(b, n) = 1, a is a quadratic residue mod n, b is a quadratic nonresidue mod n, then π(x; n, b) > π(x; n, a) occurs more often than not.

This has been proved only by assuming strong forms of the Riemann hypothesis.

The stronger conjecture of Knapowski and Turán, that the density of the numbers x for which π(x; 4, 3) > π(x; 4, 1) holds is 1 (that is, it holds for almost all x), turned out to be false.

They, however, do have a logarithmic density, which is approximately 0.9959....[1] This is for k = −4 to find the smallest prime p such that

is the Kronecker symbol), however, for a given nonzero integer k (not only k = −4), we can also find the smallest prime p satisfying this condition.

By the prime number theorem, for every nonzero integer k, there are infinitely many primes p satisfying this condition.

For positive integers k = 1, 2, 3, ..., the smallest primes p are For negative integers k = −1, −2, −3, ..., the smallest primes p are 2, 3, 608981813029, 26861, 7, 5, 2, 3, 2, 11, 5, 608981813017, 19, 3, 2, 26861, 2, 643, 11, 3, 11, 31, 2, 5, 2, 3, 608981813029, 48731, 5, 13, 2, 3, 2, 7, 11, 5, 199, 3, 2, 11, 2, 29, 53, 3, 109, 41, 2, 608981813017, 2, 3, 13, 17, 23, 5, 2, 3, 2, 1019, 5, 263, 11, 3, 2, 26861, ... (OEIS: A306500 is a subsequence, for k = −3, −4, −7, −8, −11, −15, −19, −20, −23, −24, −31, −35, −39, −40, −43, −47, −51, −52, −55, −56, −59, ... OEIS: A003657) For every (positive or negative) nonsquare integer k, there are more primes p with

Let m and n be integers such that m ≥ 0, n > 0, gcd(m, n) = 1, define a function