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 {p_1,p_2,...,p_n}" src="https://mathworld.wolfram.com/images/equations/ChebyshevBias/Inline2.gif" style="height:15px; width:93px" /> (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 {p_3,p_4,...,p_n}" src="https://mathworld.wolfram.com/images/equations/ChebyshevBias/Inline9.gif" style="height:15px; width:93px" /> (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.