Consecutive prime bias is the observed preference for certain ordered patterns of residue classes among successive primes at finite cutoffs. For example, primes greater than 5 end in the decimal digits 1, 3, 7, or 9. Successive primes are observed to repeat the same final decimal digit less frequently than a model in which these four final digits are independent and equally likely would suggest (Lemke Oliver and Soundararajan 2016, pp. E4446-E4447; Numberphile 2016).
Let 
be the 
th
prime number, let 
be an integer, and let 
and 
be integers relatively
prime to 
.
Define the ordered-pair count
 |
The cutoff applies to 
; its successor 
may exceed 
. For the first hundred million primes,
Lemke Oliver and Soundararajan (2016, p. E4446) reported approximately 4.62
million successive-prime pairs with final digits 
and 7.50 million with final digits 
, compared with about 6.25 million
for each of the 16 ordered pairs in an independent,
uniform model.
Lemke Oliver and Soundararajan (2016, p. E4447) proposed a heuristic based on the Hardy-Littlewood k-tuple conjecture.
Their conjectural model predicts, as 
,
 |
where 
is the prime counting function and 
is the totient
function, together with large secondary terms that account for the observed biases.
This concerns ordered pairs of successive primes,
whereas Chebyshev bias compares the counts of individual
primes in different residue
classes.
