Daniel Devatman Hromada, wizzion.com CEO

Subject: Practical implication of transitory nature of Nprime modulo 3 residual classes
Fellow colleagues,
with great delight have I read the article of Lemke Oliver & Soundararajan [1] which indicates an
existence of certain bias in the realm of prime numbers.
But sometimes it is the case that too much specialization disallows one to see the forest because of
the trees. Hence, authors aimed for analytic explanations there, where deeper empiric inspection of
one concrete case could potentially turn out to be equally instructive.
The case we how on our mind is that of prime modulo 3 residual classes (RCs). For 3 is the only
divisor which yields only two RCs: 1 and 2. Thus, only two possible RC transitions are possible in a
consecutive pair (CP) of primes P x and Px+1: 2 -> 1 transition and 1->2 transition. Asides this, only
two other RC combinations exist for any possible CP: 1->1 when both CP members belong to RC 1
and 2->2 when PX mod 3 == PX+1 mod 3 ==2.
Note the symmetry: two RC transitions and two "non-transition" states. Such symmetry exists, ex vi
termini, only in the realm of modulo 3.
Focusing solely on this transition / non-transition properties of consecutive pairs occurent among
first 30 million primes, one can observe:
 there are 16687076 transitions
 there are 13312923 non-transitions
 longest uninterrupted sequence of consecutive transitions consists of 32 primes
 longest uninterrupted sequence of consecutive non-transitions consists of 19 primes
 etc.
Those unafraid of induction could thus simply conjecture that given 16687076 / 13312923 =~
1.253, it is approx. 25% more probable that Px+1 will belong to different modulo3RC than PX.
In other terms: approximately 25% carbon dioxide less could be potentially emitted if machines
aiming to discover new prime PX+1 would explore:
 sequences (PX + 2 + 3*n) if it is known that PX mod 3 == 1
 sequences (PX + 4 + 3*n) if it is known that PX mod 3 == 2
It is in this point that we disagree with the expression "no practical use" contained in the statement :
"this ‘anti-sameness’ bias has no practical use or even any wider implication for number theory, as
far as Soundararajan and Lemke Oliver know" recently published in Your journal [2].
With highest regards,
Daniel D. Hromada
Berlin
[1] Robert J. Lemke Oliver, Kannan Soundararajan. Unexpected biases in the distribution of consecutive primes. 2016.
http://arxiv.org/abs/1603.03720
[2] Evelyn Lamb. Peculiar pattern found in 'random' prime numbers. 2016. http://www.nature.com/news/peculiarpattern-found-in-random-prime-numbers-1.19550
[3] https://en.wikipedia.org/wiki/Monty_Hall_problem