Unexpected biases in the distribution of consecutive primes (1603.03720v4)

Abstract: While the sequence of primes is very well distributed in the reduced residue classes (mod $q$), the distribution of pairs of consecutive primes among the permissible $\phi(q)^2$ pairs of reduced residue classes (mod $q$) is surprisingly erratic. This paper proposes a conjectural explanation for this phenomenon, based on the Hardy-Littlewood conjectures. The conjectures are then compared to numerical data, and the observed fit is very good.

• The paper identifies and analyzes unexpected biases in how consecutive prime numbers are distributed across different residue classes modulo q.
• Using conjectures, the study argues that significant secondary terms explain the observed biases in consecutive primes, deviating from expected uniform distribution.
• The findings challenge assumptions of randomness in prime distributions and have implications for number theory research and potentially cryptographic applications.

Analyzing Biases in the Distribution of Consecutive Primes

The paper by Robert J. Lemke Oliver and Kannan Soundararajan presents intriguing findings regarding the distribution of consecutive prime numbers in congruence classes modulo an integer $q$. While the distribution of individual primes in residue classes is well-established, this research explores the erratic behavior observed when examining pairs of consecutive primes, unveiling biases that contravene typical probabilistic expectations.

Main Findings and Conjectures

The paper homes in on the apparent bias in residue classes among consecutive prime pairs. This bias, which deviates from an expected uniform distribution, is tackled through a heuristic grounded in the Hardy-Littlewood prime $k$-tuples conjecture. The authors propose that while these biases are present, the primes ultimately conform to the distribution predicted by density $\frac{1}{\phi(q)^r}$ as a limit. However, substantial secondary terms in the asymptotic expressions skew the appearance of certain patterns over others. The conjectures made, therefore, address these biases and offer detailed equations to describe them.

The Main Conjecture introduced involves inequalities modulated by constants $c_1$ and $c_2$, where these biases depend significantly on whether consecutive primes repeat in the same residue class. Specifically, the biases lean heavily on the residue classes $(a_i \equiv a_{i+1} \pmod{q})$ and other specified terms.

Additionally, the main findings of the paper include several specific conjectures:

• Conjecture 1 postulates that certain consistent biases may hold indefinitely, unaffected by higher $x$.
• Conjecture 2 examines product biases between two consecutive primes preferring a quadratic non-residue characterization over a quadratic residue for a fixed odd prime $q$.
• Conjecture 3 extends these ideas to non-adjacent consecutive primes, illustrating a transition matrix paradigm that deviates from being fundamentally Markovian.

Implications and Future Prospects

The results yield profound implications for understanding how primes behave under different moduli. The persistent biases noted profoundly challenge the assumption of randomness that often accompanies discussions of prime distributions. Practically, this might influence the cryptographic domain, where prime number properties are pivotal—particularly if certain residues offer less predictability than postulated.

From a theoretical perspective, the research underscores the nuanced behavior of primes in arithmetic progressions and residue classes, rekindling interest in longstanding conjectures. The work opens avenues for further exploration into more complex interactions between primes over multiple steps and wider mathematical structures involving primes.

Numerical Validation and Observations

The authors validate their conjectures by aligning theoretical predictions with numerical data. A compelling observation emerges on $\pi(x;q,(a,b))$, illustrating how secondary terms influenced real data significantly. Their predictions showcase the sophisticated relationship between predicted densities and actual occurrences, reinforcing their conjectures' validity.

Future work may refine these numerical analyses, further investigating smaller order terms and the precise mechanisms driving these biases. As computation power and algorithmic methods continue to evolve, more definitive claims regarding the extensive landscape of bias effects in prime distributions will likely ascend, consolidating or questioning the conjectural framework laid out herein.

In summary, this paper advances a comprehensive conjectural framework to explain biases in consecutive prime distributions, laying a foundation for future mathematical inquiry and potentially impacting aspects of number theory related applications.