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Chen Prime

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A Chen prime is a prime number p for which p+2 is either a prime or semiprime. Chen primes are named after Jing Run Chen, who proved that there are infinitely many such primes (Chen 1973; Chen's theorem).

The first Chen primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... (OEIS A109611). The first primes that are not Chen primes are 43, 61, 73, 79, 97, 103, 151, ... (OEIS A102540).

The smaller member of the largest known twin-prime pair as of Oct. 2026 is the 388342-digit Chen prime 2996863034895×2^(1290000)-1 (PrimePages 2016, 2026). Excluding twin-prime cases means requiring p+2 to be a semiprime. An example in this restricted category is the 70301-digit Chen prime

(1284991359×2^(98305)+1)×(96060285×2^(135170)+1)-2

reported in Oct. 2005 (Andersen 2005, https://t5k.org/primes/page.php?id=75857).

Green and Tao (2006, p. 151) proved that there are infinitely many three-term arithmetic progressions of Chen primes. Zhou (2009, p. 302) strengthened this result by proving that the Chen primes contain arbitrarily long prime arithmetic progressions. The following 3074-digit case produces Chen primes for n=0, 1, 2, where p# denotes the primorial:

[(3850324118+892819689×n)×2411#+1]×(4787#+1)-2.

See also

Chen's Theorem, Prime Arithmetic Progression, Semiprime, Twin Primes

Portions of this entry contributed by Jens Kruse Andersen

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References

Update a linkAndersen, J. K. "Chen AP3 with 3074 digits." http://groups.yahoo.com/group/primeform/message/6381Andersen, J. K. "Chen Prime with 70301 digits." Oct. 7, 2005. https://www.worldofnumbers.com/YPFM6481.htm.Chen, J. R. "On the Representation of a Large Even Integer as the Sum of a Prime and the Product of at Most Two Primes." Scientia Sinica 16, 157-176, 1973.Evard, J.-C. "Almost Twin Primes and Chen's Theorem." http://www.math.utoledo.edu/~jevard/Page015.htm.Green, B. and Tao, T. "Restriction Theory of the Selberg Sieve, with Applications." J. Théor. nombres Bordeaux 18, 147-182, 2006. https://doi.org/10.5802/jtnb.538.PrimePages. "2996863034895×2^(1290000)-1." 2016. https://t5k.org/primes/page.php?id=122213.PrimePages. "The Top Twenty: Twin Primes." 2026. https://t5k.org/top20/page.php?id=1.Zhou, B. "The Chen Primes Contain Arbitrarily Long Arithmetic Progressions." Acta Arith. 138, 301-315, 2009. https://doi.org/10.4064/aa138-4-1.Sloane, N. J. A. Sequences A102540 and A109611 in "The On-Line Encyclopedia of Integer Sequences."

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Chen Prime

Cite this as:

Weisstein, Eric W., with contributions by Jens Kruse Andersen. "Chen Prime." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ChenPrime.html

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