Chebyshev Bias

Chebyshev noticed that the remainder upon dividing the primes by 4 gives 3 more often than 1, as plotted above in the left figure. Similarly, dividing the primes by 3 gives 2 more often than 1 (right figure). This is called the Chebyshev bias, or sometimes the prime race (Wagon 1994).

Consider the list of the first  primes  (mod 4). This list contains equal numbers of remainders 3 and 1 (mod 4) for , 3, 7, 13, 89, 2943, 2945, 2947, 2949, 2951, 2953, 50371, ... (OEIS A038691; Wagon 1994, pp. 2-3). The values of  for which the list is biased towards 1 are 2946, 50378, 50380, 50382, 50383, 50384, 50385, ... (OEIS A096628).

Defining

the values of  for which  are , 3, 7, 13, 89, 2943, 2945, 2947, ... (OEIS A038691).

Similarly, consider the list of the first  primes  (mod 3), skipping  and  since . This list contains equal numbers of remainders 2 and 1 at the values , 6, 8, 12, 14, 22, 38, 48, 50, ... (OEIS A096629). The first value of  for which the list is biased towards 1 is , as first found by Bays and Hudson in 1978 (Derbyshire 2004, p. 126), giving the first few such values as 23338590792, 23338590794, 23338590795, 23338590796, ... (OEIS A096630).

REFERENCES:

Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, pp. 125-126, 2004.

Sloane, N. J. A. Sequences A038691, A096628, A096629, and A096630 in "The On-Line Encyclopedia of Integer Sequences."

Wagon, S. The Power of Visualization. Front Range Press, 1994.