This site is supported by donations to The OEIS Foundation.

# User:Charles R Greathouse IV/Tables of special primes

## Simple Diophantine primes

These are the prime solutions to univariate Diophantine equations involving only addition, subtraction, multiplication, division, exponentiation, and the floor function. They are classified below by the rate of growth of their dominant term.

The following conjectures about primes in polynomial sequences are used in many of these types of primes:

 Single polynomial Multiple polynomials Existence / density Dirichlet's theorem (1837) / PNT in AP (1896) Dickson's conjecture (1904) / — Bunyakovsky conjecture (1857) / ※ Schinzel's hypothesis H (1958) / Bateman–Horn–Stemmler conjecture (1962)

※ The Hardy–Littlewood conjectures F and K (1923) are special cases, quadratics and ${\displaystyle x^{3}+k}$ respectively, but the general case was not handled until subsumed by the conjecture of Bateman, Horn, & Stemmler.

Linear Size A-number
Primes n infinite with density n/log n + O(n/(log n)2): prime number theorem A000040
Real Eisenstein primes 3n + 2 infinite with density 0.5n/log n + O(n/(log n)2): Dirichlet's theorem A003627
Pythagorean primes 4n + 1 infinite with density 0.5n/log n + O(n/(log n)2): Dirichlet's theorem A002144
Real Gaussian primes 4n + 3 infinite with density 0.5n/log n + O(n/(log n)2): Dirichlet's theorem A002145
Cuban primes ${\displaystyle {\tfrac {1}{4}}(3n^{2}+1)}$ conjectured infinite with density ${\displaystyle C{\sqrt {n}}/\log n}$: Hardy-Littlewood conjecture F A002407
Landau primes ${\displaystyle n^{2}+1}$ conjectured infinite with density ${\displaystyle C{\sqrt {n}}/\log n}$ (C ≈ 1.372813): Hardy-Littlewood conjecture E A002496
Central polygonal primes ${\displaystyle n^{2}-n+1}$ conjectured infinite with density ${\displaystyle C{\sqrt {n}}/\log n}$: Hardy-Littlewood conjecture F A002383
Centered triangular primes ${\displaystyle {\tfrac {1}{2}}(3n^{2}+3n+2)}$ conjectured infinite: Schinzel's hypothesis H A125602
Centered square primes ${\displaystyle {\tfrac {1}{2}}(4n^{2}+4n+2)}$ conjectured infinite with density ${\displaystyle C{\sqrt {n}}/\log n}$: Hardy-Littlewood conjecture F A027862
Centered pentagonal primes ${\displaystyle {\tfrac {1}{2}}(5n^{2}+5n+2)}$ conjectured infinite: Schinzel's hypothesis H A145838
Centered hexagonal primes ${\displaystyle {\tfrac {1}{2}}(6n^{2}+6n+2)}$ conjectured infinite with density ${\displaystyle C{\sqrt {n}}/\log n}$: Hardy-Littlewood conjecture F A002407
Cuban primes (variant) ${\displaystyle 3n^{2}+1}$ conjectured infinite with density ${\displaystyle C{\sqrt {n}}/\log n}$: Hardy-Littlewood conjecture F A002648
Centered heptagonal primes ${\displaystyle {\tfrac {1}{2}}(7n^{2}+7n+2)}$ conjectured infinite: Schinzel's hypothesis H A144974
Centered decagonal primes ${\displaystyle {\tfrac {1}{2}}(10n^{2}+10n+2)}$ conjectured infinite with density ${\displaystyle C{\sqrt {n}}/\log n}$: Hardy-Littlewood conjecture F A090562
Star primes ${\displaystyle 6n^{2}-6n+1}$ conjectured infinite with density ${\displaystyle C{\sqrt {n}}/\log n}$: Hardy-Littlewood conjecture F A083577
Cubic Size A-number
${\displaystyle n^{3}+2}$ conjectured infinite with density ${\displaystyle C{\sqrt[{3}]{n}}/\log n}$ (C ≈ 1.2985): Hardy-Littlewood conjecture K A144953
Exponential Size A-number
Wagstaff primes ${\displaystyle {\tfrac {1}{3}}(2^{n}+1)}$ unknown, trivial density O(log n) A000979
Mersenne primes ${\displaystyle 2^{n}-1\,}$ conjectured infinite with density log2 log n + o(log log n): Lenstra–Pomerance–Wagstaff conjecture A000668
Thābit primes ${\displaystyle 3\cdot 2^{n}-1}$ unknown, trivial density O(log n) A007505
Cullen primes ${\displaystyle n\cdot 2^{n}+1}$ unknown, density o(log n)[1] A050920
Woodall primes ${\displaystyle n\cdot 2^{n}-1}$ unknown, density o(log n)[2] A050918
Super-exponential Size A-number
Double Mersenne primes ${\displaystyle {\tfrac {1}{2}}\cdot 2^{2^{n}}-1}$ unknown, trivial density O(log log n/log log log n)
Legendre primes ${\displaystyle \lfloor A^{2^{n}}\rfloor }$ for ${\displaystyle A=1.524\ldots }$ infinite with density Θ(log log n) under Legendre's conjecture; otherwise possibly ill-defined A059784
Fermat primes ${\displaystyle 2^{2^{n}}+1}$ conjectured finite (5 known) A019434
Mills primes ${\displaystyle \lfloor A^{3^{n}}\rfloor }$ for ${\displaystyle A=1.306\ldots }$ infinite by construction[3] with density log3 log n + O(1) A051254

## Other Diophantine primes

These are the prime solutions to more complicated Diophantine equations.

Size A-number
Alternating factorial prime ${\displaystyle P=n!-(n-1)!+\cdots \pm 1!}$ finite[4] A071828
Bertrand primes ${\displaystyle \lfloor 2^{b}\rfloor ,\lfloor 2^{2^{b}}\rfloor ,\ldots }$ for ${\displaystyle b=1.251\ldots }$ infinite by Bertrand's postulate with density lg  n + O(1) A051501
Cuban primes (generalized) ${\displaystyle P=x^{2}+3y^{2}}$ infinite with density 0.5n/log n + O(n/log2 n): Dirichlet's theorem A007645
Euclid primes ${\displaystyle P=n\#+1}$ conjectured infinite with density eγ log n[5] A018239
Factorial primes ${\displaystyle P=n!-1\,}$ conjectured infinite with density eγ log n[5] A055490
Factorial primes ${\displaystyle P=n!+1\,}$ conjectured infinite with density eγ log n[5] A088054
Fouvry–Iwaniec primes ${\displaystyle P=x^{2}+p^{2}}$ infinite with density Θ(n/(log n)2)[6] A185086
Friedlander–Iwaniec primes ${\displaystyle P=m^{2}+n^{4}}$ Θ(n3/4/log n)[7] A028916
Half-octavan primes ${\displaystyle P=(m^{8}+n^{8})/2}$ trivial density O(n1/4), conjectured infinite A290780
Half-quartan primes ${\displaystyle P=(m^{4}+n^{4})/2}$ trivial density O(n1/2), conjectured infinite A002646
Heath-Brown primes ${\displaystyle P=m^{3}+2n^{3}}$ Θ(n2/3/log n)[8] A173587
Heath-Brown–Li primes ${\displaystyle P=n^{2}+p^{4}\,}$ Θ(n3/4/log2 n)[9] A281792
Leyland primes ${\displaystyle P=m^{n}+n^{m}\,}$ (m, n > 1) trivial density O((log n)2log log n) A094133
Markov primes ${\displaystyle m^{2}+n^{2}+P^{2}=3mnP}$ trivial density O((log n)2)[10] A178444
Octavan primes ${\displaystyle P=m^{8}+n^{8}}$ trivial density O(n1/4), conjectured infinite A006686
Pierpont primes ${\displaystyle P=2^{m}3^{n}+1}$ conjectured infinite[11] with density Θ(log n), trivial density O(log2 n) A005109
Primorial primes ${\displaystyle P=n\#-1}$ conjectured infinite with density eγ log n[5] A057705
Primorial primes ${\displaystyle P=n\#+1}$ conjectured infinite with density eγ log n[5] A005234
Proth primes ${\displaystyle P=n\cdot 2^{m}+1}$ with ${\displaystyle n<2^{m}}$ trivial density O(n1/2) A080076
Quartan primes ${\displaystyle P=m^{4}+n^{4}}$ trivial density O(n1/2), conjectured infinite A002645
Semi-octavan primes ${\displaystyle P=m^{4}+n^{8}}$ trivial density O(n3/8), conjectured infinite A291206
Solinas primes ${\displaystyle P=2^{m}\pm 2^{n}\pm 1}$ with ${\displaystyle 0 trivial density O(log2 n) A165255
Soundararajan primes ${\displaystyle P=1^{1}+2^{2}+\cdots +n^{n}}$ density O(log n/(log log n)2)[12] A073826
Three-square primes ${\displaystyle P=l^{2}+m^{2}+n^{2}}$ 0.75n/log n + O(n/(log n)2) A042998
Two-square primes ${\displaystyle P=m^{2}+n^{2}}$ 0.5n/log n + O(n/(log n)2): Fermat's theorem on sums of two squares A002313

## Recurrence relation primes

These recurrence relations are exponential, and so all of these sequences (and thus their prime subsets) are trivially of density O(log n).

Size A-number
Fibonacci primes ${\displaystyle a_{n}=a_{n-1}+a_{n-2}}$ with ${\displaystyle a_{1}=a_{2}=1}$ density O(log n/log log n) A005478
Lucas primes ${\displaystyle a_{n}=a_{n-1}+a_{n-2}}$ with ${\displaystyle a_{1}=1,a_{2}=3}$ density O(log n/log log n) A005479
Padovan primes ${\displaystyle a_{n}=a_{n-2}+a_{n-3}}$ with ${\displaystyle a_{0}=a_{1}=a_{2}=1}$ trivial density O(log n) A100891
NSW primes ${\displaystyle a_{n}=2a_{n-1}+a_{n-2}}$ with ${\displaystyle a_{0}=a_{1}=1}$ trivial density O(log n) A088165
Pell primes ${\displaystyle a_{n}=2a_{n-1}+a_{n-2}}$ with ${\displaystyle a_{1}=1,a_{2}=2}$ trivial density O(log n) A086383
Perrin primes ${\displaystyle a_{n}=a_{n-2}+a_{n-3}}$ with ${\displaystyle a_{1}=0,a_{1}=2,a_{2}=3}$ trivial density O(log n) A074788

## Prime constellations

Dickson's conjecture implies that each admissible prime k-tuple has an infinite number of primes. Their density, by the Bateman-Horn-Stemmler conjecture, is ${\displaystyle cn/(\log n)^{k}}$, where c is an effectively computable constant depending only on the form of the constellation.

Size A-number
Twin primes ${\displaystyle (p,p+2)}$ density O(n (log log n)2/(log n)2): Brun's theorem; conjectured infinite (twin prime conjecture) with density 2C2n/(log n)2 A001359
Cousin primes ${\displaystyle (p,p+4)}$ conjectured infinite with density 2C2n/(log n)2 A023200
Sexy primes ${\displaystyle (p,p+6)}$ conjectured infinite with density 4C2n/(log n)2 A023201
Prime triplets ${\displaystyle (p,p+2,p+6)}$ conjectured infinite with density 4.5C3n/(log n)3 A022004
Prime triplets ${\displaystyle (p,p+4,p+6)}$ conjectured infinite with density 4.5C3n/(log n)3 A022005
Prime quadruplets ${\displaystyle (p,p+2,p+6,p+8)}$ conjectured infinite with density 13.5C4n/(log n)4 A007530

## Primes by size

Size A-number
Odd primes ${\displaystyle p>2}$ infinite with density n/log n + O(n/(log n)2): prime number theorem A065091
Titanic primes ${\displaystyle p>10^{999}}$ infinite with density n/log n + O(n/(log n)2): prime number theorem A074282+10999
Gigantic primes ${\displaystyle p>10^{9999}}$ infinite with density n/log n + O(n/(log n)2): prime number theorem A142587+109999
Megaprimes ${\displaystyle p>10^{999999}}$ infinite with density n/log n + O(n/(log n)2): prime number theorem

## Base-dependent primes

Size (base 10) A-number
Additive primes infinite with conjectured density ~ 1.5n/log n log log n[13], some unconditional bounds are known A046704
Circular primes conjectured to be the repunit primes, plus a finite number of other primes A016114
Dihedral primes density O((n/log n)0.699) by the normality of the Copeland–Erdős constant (only 5 digits can be used, and log105 < 0.699) A038136
Emirps conjectured logarithmic density Θ(n/(log n)2)1 A006567
Friedman primes Θ(n/log n) since at least one residue class mod 108 is always a Friedman number A112419
Full reptend primes conjectured infinite with density CArtinn/log n: Artin's conjecture on primitive roots A001913
Happy primes conjectured infinite with density Θ(n/log n) A035497
Left-truncatable primes finite (4260 elements) A024785
Minimal primes finite (26 elements) A071062
Palindromic primes density O(n1/2log log log n/log log n)[14], conjectured infinite A002385
Pandigital primes infinite with density n/log n + O(n/(log n)2); relative complement has density O(n0.955). A050288
Permutable primes conjectured to be the repunit primes, plus a finite number of other primes; density O((log n)2)[15] A003459
Primeval primes trivially O((log n)10) because digits are nondecreasing A119535
Repunit primes conjectured infinite with density Θ(log log n),[16] trivial density O(log n/log log n) since the length must be prime A004022
Right-truncatable primes finite (83 elements) A024770
Self primes unknown A006378
Short period primes conjectured infinite with density (1-CArtin)n/log n: Artin's conjecture on primitive roots A006559
Smarandache–Wellin prime unknown, trivial density O(log n/log log n) A069151
Strobogrammatic primes density O((n/log n)0.699) by the normality of the Copeland–Erdős constant (only 5 digits can be used, and log105 < 0.699) A069151
Unique primes trivial density O(log n), conjectured infinite A040017
Weakly prime numbers infinite with density Θ(n/log n)[17] A050249

## Other classes of primes

Size A-number
Asymmetric primes density n/log n + O(n/log n)[18] A090191
Balanced primes conjectured infinite A006562
Bell primes trivial density O(log n/log log n); Pratt[19] conjectures that they are infinite A051131
Bertrand primes density lg n + O(1) A006992
Chen primes infinite[20] with density O(n log log n/(log n)2)? A109611
Elite primes density O(n/(log n)2),[21] conjectured infinite with density O((log n)c) for c ≥ 1[22] A102742
Flat primes infinite with density 2An/log n + o(n/log n)[23] A192862
Fortunate primes conjectured infinite; probably of positive relative density; under Fortune's conjecture, density Ω(log  n) A046066
Good primes infinite[24] A028388
Green–Tao prime infinite with density Ω(log log log log log log log n)[25][26] and O(log n); conjectured roughly 2 log n/log log n[27] A005115
Harmonic primes conjectured infinite with density (1/e)n/log n + o(n/log n)[28] A092101
Higgs primes conjectured infinite with density o(n/log n)[29] A007459
Highly cototient primes unknown A105440
Irregular primes infinite[30] with density Ω(log log n/log log log n);[31] conjectured density (1-e-1/2)n/log n A000928
Lucky primes unknown A031157
Ménage primes heuristically log log log n[32] A249510
Mirimanoff primes heuristically log log n? A014127
Motzkin primes trivial density O(log n) A092832
Partition primes trivial density O((log n)2) A049575
Pillai primes infinite[33] A063980
Ramanujan primes infinite[34] with density 0.5n/log n + o(n/log n)[35] A104272
Regular primes conjectured infinite with density e-1/2n/log n A007703
Safe primes conjectured infinite with density C2n/2(log n/2)2 + o(n/(log n)2) A005385
Sophie Germain primes density O(n/(log n)2); conjectured infinite with density 2C2n/(log n)2 A005384
Stern primes conjectured finite (8 known) A042978
Super-primes infinite, density n/(log n)2 + O(n log log n/(log n)3)[36] A006450
Subfactorial primes conjectured infinite, trivial density O(log n/log log n) A100015
Supersingular primes finite (15 elements) A002267
Sylvester primes infinite with density Ω(log log n)[37] and O(n/log n log log log n)[38] A007996
Symmetric primes density O(n/(log n)1.027),[18] conjectured infinite A090190
Thin primes density O(n/(log n)2)[23], conjectured density Θ(n/(log n)2)[23] A192869
Ulam primes conjectured density Θ(n/log n)? A068820
Wall–Sun–Sun (or Fibonacci–Wieferich) primes heuristically infinite with density roughly log log n,[39][40][41] none known
Wedderburn-Etherington prime trivial density O(log n) A136402
Wieferich primes conjectured infinite with density Θ(log log n)[42][43] A001220
Wilson primes heuristic density Θ(log log n)[42] A007540
Wolstenholme primes conjectured infinite with density about log log n[44] A088164
Zhou primes infinite[45] A291525

## References

1. C. Hooley (1976). Applications of Sieve Methods to the Theory of Numbers. Cambridge University Press. p. 119.
2. Hiromi Suyama, cited in Wilfrid Keller (1995). “New Cullen primes”. Mathematics of Computation 64: pp. 1733–1741.
3. W. H. Mills (1947). “A prime-representing function”. Bulletin of the American Mathematical Society 53: p. 604.
4. Miodrag Živković (1999). “The number of primes ${\displaystyle \sum _{i=1}^{n}(-1)^{n-i}i!}$ is finite”. Mathematics of Computation 68: pp. 403–409.
5. Chris K. Caldwell; Yves Gallot (2002). “On the primality of n! ± 1 and 2 × 3 × 5 × … × p ± 1”. Mathematics of Computation 71 (237): pp. 441–448.
6. Étienne Fouvry and Henryk Iwaniec (1997). “Gaussian primes”. Acta Arithmetica 79 (3): pp. 249–287. .
7. John Friedlander; Henryk Iwaniec (1997). “Using a parity-sensitive sieve to count prime values of a polynomial”. Proceedings of the National Academy of Sciences 94: pp. 1054–1058.
8. D. R. Heath-Brown (2001). “Primes represented by x3 + 2y3. Acta Mathematica 186: pp. 1–84.
9. D. R. Heath-Brown; Xiannan Li (2017). “Prime values of ${\displaystyle a^{2}+p^{4}}$. Inventiones Mathematicae 208 (2): pp. 1–59.
10. Don Zagier (1982). “On the number of Markoff numbers below a given bound”. Mathematics of Computation 39 (160): pp. 709–723.
11. Andrew M. Gleason (1988). “Angle trisection, the heptagon, and the triskaidecagon”. The American Mathematical Monthly 95 (3): pp. 185–194.
12. K. Soundararajan (1993). “Primes in a sparse sequence”. Journal of Number Theory 43 (2): pp. 220–227.
13. Glyn Harman (2012). “Counting primes whose sum of digits is prime”. Journal of Integer Sequences 15: pp. 7 pp.
14. William D. Banks; Derrick N. Hart; Mayumi Sakata (2004). “Almost all palindromes are composite”. Mathematical Research Letters 11: pp. 853–868.
15. Dmitry Mavlo (1995). “Absolute prime numbers”. The Mathematical Gazette 79 (485): pp. 299–304.
16. Chris Caldwell, Repunit on the Prime Pages
17. Terence Tao (2011). “A remark on primality testing and decimal expansions”. Journal of the Australian Mathematical Society 91 (3): pp. 405–413. arXiv:0802.3361.
18. Peter Fletcher; William Lindgren; Carl Pomerance (1996). “Symmetric and asymmetric primes”. Journal of Number Theory 58 (1): pp. 89–99.
19. Vaughan Pratt, cited in Martin Gardner (1992). "Chapter 2: The Tinkly Temple Bells". Fractal Music, Hypercards, and more...: Mathematical Recreations from Scientific American. W. H. Freeman. pp. 30-35.
20. J. R. Chen (1973). “On the representation of a larger even integer as the sum of a prime and the product of at most two primes”. Scientia Sinica 16: pp. 157–176.
21. M. Křížek; F. Luca; L. Somer (2002). “On the convergence of series of reciprocals of primes related to the Fermat numbers”. Journal of Number Theory 97: pp. 95–112.
22. Tom Müller (2006). “Searching for large elite primes”. Experimental Mathematics 15 (2): pp. 183–186.
23. Kevin Broughan and Zhou Qizhi (2010). “Flat primes and thin primes”. Bulletin of the Australian Mathematical Society 82 (2): pp. 282–292.
24. Carl Pomerance (1979). “The prime number graph”. Mathematics of Computation 33 (145): pp. 399–408.
25. Ben Green; Terence Tao (2008). “The primes contain arbitrarily long arithmetic progressions”. Annals of Mathematics 167 (2): pp. 481–547. arXiv:math/0404188.
26. Ben Green and Terence Tao, A bound for progressions of length k in the primes
27. Andrew Granville (2008). “Prime number patterns”. Amer. Math. Monthly 115 (4): pp. 279–296.
28. David W. Boyd (1994). “A p-adic study of the partial sums of the harmonic series”. Experimental Mathematics 3 (4): pp. 287–302. .
29. Stanley N. Burris and Karen A. Yeats (2005). “The saga of the High School Identities”. Algebra Universalis 52 (2–3): pp. 325–342. doi:10.1007/s00012-004-1900-2. .
30. K. L. Jensen (1915). “Om talteoretiske Egenskaber ved de Bernoulliske Tal”. Nyt Tidsskrift für Matematik Afdeling B 28: pp. 73–83.
31. Florian Luca, Amalia Pizarro-Madariaga, and Carl Pomerance (2015). “On the counting function of irregular primes”. Indag. Math. (N.S.) 26 (1): pp. 147–161.
32. Adam P. Goucher, Ménage primes (2014)
33. G. E. Hardy; M. V. Subbarao (2002). “A modified problem of Pillai and some related questions”. American Mathematical Monthly 109 (6): pp. 554–559.
34. Srinivas Ramanujan (1919). “A proof of Bertrand's postulate”. Journal of the Indian Mathematical Society 11: pp. 181–182.
35. Jonathan Sondow (2009). “Ramanujan Primes and Bertrand's Postulate”. American Mathematical Monthly 116: pp. 630–635. arXiv:0907.5232.
36. Kevin A. Broughan; A. Ross Barnett (2009). “On the subsequence of primes having prime subscripts”. Journal of Integer Sequences 12. article 09.2.3.
37. Rafe Jones (2008). “The density of prime divisors in the arithmetic dynamics of quadratic polynomials”. Journal of the London Mathematical Society 78 (2): pp. 523–544. arXiv:math/0612415.
38. R. W. K. Odoni (1985). “On the prime divisors of the sequence ${\displaystyle \scriptstyle w_{n+1}=1+w_{1}...w_{n}}$”. Journal of the London Mathematical Society 32 (1): pp. 1–11.
39. Richard J. McIntosh and Eric L. Roettger (2007). “A search for Fibonacci–Wieferich and Wolstenholme primes”. Mathematics of Computation 76: pp. 2087–2094.
40. Jiří Klaška (2008). “Short remark on Fibonacci-Wieferich primes”. Acta Mathematica et Informatica Universitatis Ostraviensis 15 (15): pp. 21–25.
41. Grell, George; Peng, Wayne (2015). "Wall's Conjecture and the ABC Conjecture". arΧiv:1511.01210.
42. R. Crandall; K. Dilcher; C. Pomerance (1997). “A search for Wieferich and Wilson primes”. Mathematics of Computation 66 (217): pp. 433–449.
43. Joshua Knauer and Jörg Richstein (2005). “The continuing search for Wieferich primes”. Mathematics of Computation 74: pp. 1559–1563.
44. R. J. McIntosh (1995). “On the converse of Wolstenholme's Theorem”. Acta Arithmetica 71: pp. 381–389.
45. Binbin Zhou (2009). “The Chen primes contain arbitrarily long arithmetic progressions”. Acta Arithmetica 138: pp. 301–315.