Draft:Equidistant prime pair

• Comment: A cut-off lead in the beginning, lack of reliable sources to support its notability, and some incomprehensible mathematical symbols which lead to technical (e.g. uneasy understanding) for readers. Most of the sources are from websites, especially MathWorld. Are there more reliable sources (see WP:RS) mentions more about equidistant primes? Dedhert.Jr (talk) 09:22, 15 March 2025 (UTC)

Goldbach's prime triangle: Central column showing primes (where the y-axis represents n), with equidistant prime pairs shown on the x-axis as n increases (for integers n>2)

Prime numbers equidistant from n (for integers > 2) and primes in column n

Primes equidistant from n = 12

An equidistant prime pair is a pair of prime numbers (${\displaystyle p_{1},p_{2}}$) that have the same distance ${\displaystyle d\geq 0}$ from an integer n, such that ${\displaystyle n-d=p_{1}}$ and ${\displaystyle n+d=p_{2}}$. Equidistant prime pairs are also Goldbach partitions.^[1][2][3] Primes with such properties have been described by several authors, such as Richard Crandall, Carl B. Pomerance^[4], Leon Ehrenpreis^[5] and others. Since equidistant prime pairs illustrate different ways of how even integers can be written as the sum of two primes, they can also be linked to Goldbach's conjecture.^[6] According to Oliveira e Silva, Herzog and Pardi, the additive decomposition ${\displaystyle n=p+q}$ is called a Goldbach partition of n, where n is an even integer larger than four, and p and q are odd prime numbers.^[7] Equidistant prime pairs are based on the same idea, but include the additional distance variable ${\displaystyle d}$.

By definition, every prime itself can also be considered to be an equidistant prime pair, since the distance from n is zero (${\displaystyle n-0}$ and ${\displaystyle n+0}$, respectively). As n gets larger, the number of prime pairs that sum to an even integer generally increases, as indicated by the Goldbach partition function.^[8][9]

Twin primes can be expressed as an equidistant prime pair of the form ${\displaystyle n-1}$ and ${\displaystyle n+1}$ (where the distance ${\displaystyle d=1}$). From this perspective, equidistant prime pairs could be seen as a more generalized form of twin primes.

The number of equidistant prime pairs for integers n > 0 corresponds to OEIS sequence A045917.^[10]

Equidistant prime pairs mentioned in this article are not to be confused with sequences of primes that are equidistant to eachother, such as Primes in arithmetic progressions according to the Green-Tao theorem.

Visualizing equidistant prime pairs

Equidistant prime pairs for each n can be visualized through Goldbach's Prime Triangle, a plot with a central column representing n (y-axis), where additional prime pairs are shown on the x-axis as n increases. The basic pattern of the triangle also emerges when applying an alternative method introduced in 2012 by Cunningham and Ringland.

See also

• Goldbach's conjecture
• List of prime numbers
• Twin prime

References

1. Winkelmann, Jörg (28 February 2025). "Goldbach's Prime Triangle — A Recreational Math Journey with an Introduction to Equidistant Primes". Zenodo. Retrieved 2 March 2025.
2. Stewart, John (2004). "Goldbach's Conjecture in non-integer contexts" (PDF). Library and Archives Canada. p. 76. Archived (PDF) from the original on March 16, 2025. Retrieved March 16, 2025.
3. Ricardez, Javier Horacio Perez (2025). "Conjecture on the Sum of Primes in Contiguous Intervals". ResearchGate (Preprint). doi:10.13140/RG.2.2.15288.05124.
4. Crandall, Richard E.; Pomerance, Carl (2005). Prime numbers: a computational perspective. Lecture notes in statistics (Springer-Verlag) (2nd ed.). New York, NY: Springer. p. 345. ISBN 978-0-387-28979-3.
5. Farkas, Hershel M.; Ehrenpreis, Leon, eds. (2012). From fourier analysis and number theory to radon transforms and geometry: in memory of Leon Ehrenpreis. Developments in mathematics. New York ; London: Springer. p. 225. ISBN 978-1-4614-4074-1. OCLC 793689419.
6. Hardy, G. H.; Littlewood, J. E. (1923-12-01). "Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes". Acta Mathematica. 44 (1): 1–70. doi:10.1007/BF02403921. ISSN 1871-2509.
7. Oliveira e Silva, Tomás; Herzog, Siegfried; Pardi, Silvio (2014). "Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4⋅10¹⁸". Mathematics of Computation. 83 (288): 2033–2060. doi:10.1090/S0025-5718-2013-02787-1. ISSN 0025-5718.
8. Weisstein, Eric W. (27 February 2025). "Goldbach Partition". From MathWorld--A Wolfram Web Resource. Retrieved 2 March 2025.
9. Zhang, Yitang (2014). "Bounded gaps between primes | Annals of Mathematics". Annals of Mathematics. 179 (3): 1121–1174. doi:10.4007/annals.2014.179.3.7.
10. "From Goldbach problem: number of decompositions of 2n into unordered sums of two primes". OEIS. 16 February 2025. Retrieved 2 March 2025.

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Categories:

• Prime numbers
• Classes of prime numbers