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).
