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User:Charles R Greathouse IV/Tables of special primes
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Diophantine primes
The following conjectures about primes in polynomial sequences are used in many of these types of primes:
Existence / density | Single polynomial | Multiple polynomials |
---|---|---|
Linear | Dirichlet's theorem (1837) / PNT in AP (1896) | Dickson's conjecture (1904) / — |
Arbitrary degree | 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 respectively, but the general case was not handled until subsumed by the conjecture of Bateman, Horn, & Stemmler.
Simple univariate 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.
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 |
Quadratic | Size | A-number |
Cuban primes | conjectured infinite with density : Hardy-Littlewood conjecture F | A002407 |
Landau primes | conjectured infinite with density (C ≈ 1.372813): Hardy-Littlewood conjecture E | A002496 |
Central polygonal primes | conjectured infinite with density : Hardy-Littlewood conjecture F | A002383 |
Centered triangular primes | conjectured infinite with density : Hardy-Littlewood conjecture F | A125602 |
Centered square primes | conjectured infinite with density : Hardy-Littlewood conjecture F | A027862 |
Centered pentagonal primes | conjectured infinite with density : Hardy-Littlewood conjecture F | A145838 |
Centered hexagonal primes | conjectured infinite with density : Hardy-Littlewood conjecture F | A002407 |
Cuban primes (variant) | conjectured infinite with density : Hardy-Littlewood conjecture F | A002648 |
Centered heptagonal primes | conjectured infinite with density : Hardy-Littlewood conjecture F | A144974 |
Centered decagonal primes | conjectured infinite with density : Hardy-Littlewood conjecture F | A090562 |
Star primes | conjectured infinite with density : Hardy-Littlewood conjecture F | A083577 |
Cubic | Size | A-number |
conjectured infinite with density (C ≈ 1.2985): Hardy-Littlewood conjecture K | A144953 | |
Exponential | Size | A-number |
Wagstaff primes | unknown, trivial density O(log n) | A000979 |
Mersenne primes | conjectured infinite with density log2 log n + o(log log n): Lenstra–Pomerance–Wagstaff conjecture | A000668 |
Thābit primes | unknown, trivial density O(log n) | A007505 |
Cullen primes | unknown, density o(log n)[1] | A050920 |
Woodall primes | unknown, density o(log n)[2] | A050918 |
Super-exponential | Size | A-number |
Double Mersenne primes | unknown, trivial density O(log log n/log log log n) | |
Legendre primes for | infinite with density Θ(log log n) under Legendre's conjecture; otherwise possibly ill-defined | A059784 |
Fermat primes | conjectured finite (5 known) | A019434 |
Mills primes for | infinite by construction[3] with density log3 log n + O(1) | A051254 |
Primes in quadratic bivariate forms
Discriminant | Size | A-number | Definiteness | |
---|---|---|---|---|
Generalized cuban primes | -3 | infinite with density 0.5n/log n + O(n/log2 n): Dirichlet's theorem | A007645 | Positive definite |
-12 | ||||
-4 | 0.5n/log n + O(n/(log n)2): Fermat's theorem on sums of two squares | A002313 | Positive definite | |
(excluding 2) | -16 | |||
-7 | A106856 | Positive definite | ||
-132 | infinite with density 0.15n/log n + O(n/log2 n): Dirichlet's theorem | A139827 | Positive definite | |
9 | infinite with density 0.5n/log n + O(n/log2 n): Dirichlet's theorem | A002476 | Indefinite |
Simple bivariate Diophantine primes
These are the prime solutions to bivariate Diophantine equations involving only addition, subtraction, multiplication, division, exponentiation, and the floor function, excluding those in quadratic bivariate forms which are covered above.
Size | A-number | |
---|---|---|
Friedlander–Iwaniec primes | Θ(n3/4/log n)[4] | A028916 |
Half-octavan primes | trivial density O(n1/4), conjectured infinite | A290780 |
Half-quartan primes | trivial density O(n1/2), conjectured infinite | A002646 |
Heath-Brown primes | Θ(n2/3/log n)[5] | A173587 |
Leyland primes (m, n > 1) | trivial density O((log n)2log log n) | A094133 |
Octavan primes | trivial density O(n1/4), conjectured infinite | A006686 |
Pierpont primes | conjectured infinite[6] with density Θ(log n), trivial density O(log2 n) | A005109 |
Proth primes with | trivial density O(n1/2) | A080076 |
Quartan primes | trivial density O(n1/2), conjectured infinite | A002645 |
Semi-octavan primes | trivial density O(n3/8), conjectured infinite | A291206 |
Solinas primes with | trivial density O(log2 n) | A165255 |
Other Diophantine primes
These are the prime solutions to more complicated Diophantine equations.
Size | A-number | |
---|---|---|
Alternating factorial prime | finite[7] | A071828 |
Bertrand primes for | infinite by Bertrand's postulate with density lg ⁎ n + O(1) | A051501 |
Euclid primes | conjectured infinite with density eγ log n[8] | A018239 |
Factorial primes | conjectured infinite with density eγ log n[8] | A055490 |
Factorial primes | conjectured infinite with density eγ log n[8] | A088054 |
Fouvry–Iwaniec primes | infinite with density Θ(n/(log n)2)[9] | A185086 |
Heath-Brown–Li primes | Θ(n3/4/log2 n)[10] | A281792 |
Markov primes | trivial density O((log n)2)[11] | A178444 |
Primorial primes | conjectured infinite with density eγ log n[8] | A057705 |
Primorial primes | conjectured infinite with density eγ log n[8] | A005234 |
Soundararajan primes | density O(log n/(log log n)2)[12] | A073826 |
Three-square primes | 0.75n/log n + O(n/(log n)2) | A042998 |
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 with | density O(log n/log log n) | A005478 |
Lucas primes with | density O(log n/log log n) | A005479 |
Padovan primes with | trivial density O(log n) | A100891 |
NSW primes with | density O(log n/log log n)[13] | A088165 |
Pell primes with | density O(log n/log log n) | A086383 |
Perrin primes with | 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 , where c is an effectively computable constant depending only on the form of the constellation.
Size | A-number | |
---|---|---|
Twin primes | 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 | conjectured infinite with density 2C2n/(log n)2 | A023200 |
Sexy primes | conjectured infinite with density 4C2n/(log n)2 | A023201 |
Prime triplets | conjectured infinite with density 4.5C3n/(log n)3 | A022004 |
Prime triplets | conjectured infinite with density 4.5C3n/(log n)3 | A022005 |
Prime quadruplets | conjectured infinite with density 13.5C4n/(log n)4 | A007530 |
Primes by size
Size | A-number | |
---|---|---|
Odd primes | infinite with density n/log n + O(n/(log n)2): prime number theorem | A065091 |
Titanic primes | infinite with density n/log n + O(n/(log n)2): prime number theorem | A074282+10999 |
Gigantic primes | infinite with density n/log n + O(n/(log n)2): prime number theorem | A142587+109999 |
Megaprimes | 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[14], 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) | A134996 |
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)[15], 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)[16] | A003459 |
Primeval primes | trivially O((log n)10) because digits are nondecreasing | A119535 |
Repunit primes | conjectured infinite with density Θ(log log n),[17] 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)[18] | A050249 |
Other classes of primes
Size | A-number | |
---|---|---|
Asymmetric primes | density n/log n + o(n/log1.086 n)[19] | A090191 |
Balanced primes | conjectured infinite | A006562 |
Bell primes | trivial density O(log n/log log n); Pratt[20] conjectures that they are infinite | A051131 |
Bertrand primes | density lg n + O(1) | A006992 |
Chen primes | infinite[21] with density O(n log log n/(log n)2)? | A109611 |
Elite primes | density O(n/(log n)3/2),[22][23] O(n5/6) on GRH,[23] conjectured infinite with density O((log n)c) for c ≥ 1[24] | A102742 |
Flat primes | infinite with density 2An/log n + o(n/log n)[25] | A192862 |
Fortunate primes | conjectured infinite; probably of positive relative density; under Fortune's conjecture, density Ω(log ⁎ n) | A046066 |
Good primes | infinite[26] | A028388 |
Green–Tao prime | infinite with density Ω(log log log log log log log n)[27][28] and O(log n); conjectured roughly 2 log n/log log n[29] | A005115 |
Harmonic primes | conjectured infinite with density (1/e)n/log n + o(n/log n)[30] | A092101 |
Higgs primes | conjectured infinite with density o(n/log n)[31] | A007459 |
Highly cototient primes | unknown | A105440 |
Irregular primes | infinite[32] with density Ω(log log n/log log log n);[33] conjectured density (1-e-1/2)n/log n | A000928 |
Lucky primes | unknown | A031157 |
Ménage primes | heuristically log log log n[34] | 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[35] | A063980 |
Ramanujan primes | infinite[36] with density 0.5n/log n + o(n/log n)[37] | 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)[38] | 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)[39] and O(n/log n log log log n)[40] | A007996 |
Symmetric primes | density n(log log n)O(1)/(log n)1.086...,[41] conjectured infinite | A090190 |
Thin primes | density O(n/(log n)2)[25], conjectured density Θ(n/(log n)2)[25] | A192869 |
Ulam primes | conjectured density Θ(n/log n)? | A068820 |
Wall–Sun–Sun primes | heuristically infinite with density roughly log log n,[42][43][44] none known | |
Wedderburn-Etherington prime | trivial density O(log n) | A136402 |
Wieferich primes | conjectured infinite with density Θ(log log n)[45][46] | A001220 |
Wilson primes | heuristic density Θ(log log n)[45] | A007540 |
Wolstenholme primes | conjectured infinite with density about log log n[47] | A088164 |
Zhou primes | infinite[48] | A291525 |
References
- ↑ C. Hooley (1976). Applications of Sieve Methods to the Theory of Numbers. Cambridge University Press. p. 119.
- ↑ Hiromi Suyama, cited in Wilfrid Keller (1995). “New Cullen primes”. Mathematics of Computation 64: pp. 1733–1741 .
- ↑ W. H. Mills (1947). “A prime-representing function”. Bulletin of the American Mathematical Society 53: p. 604 .
- ↑ 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 .
- ↑ D. R. Heath-Brown (2001). “Primes represented by x3 + 2y3”. Acta Mathematica 186: pp. 1–84 .
- ↑ Andrew M. Gleason (1988). “Angle trisection, the heptagon, and the triskaidecagon”. The American Mathematical Monthly 95 (3): pp. 185–194 .
- ↑ Miodrag Živković (1999). “The number of primes is finite”. Mathematics of Computation 68: pp. 403–409 .
- ↑ 8.0 8.1 8.2 8.3 8.4 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 .
- ↑ Étienne Fouvry and Henryk Iwaniec (1997). “Gaussian primes”. Acta Arithmetica 79 (3): pp. 249–287 ..
- ↑ D. R. Heath-Brown; Xiannan Li (2017). “Prime values of ”. Inventiones Mathematicae 208 (2): pp. 1–59 .
- ↑ Don Zagier (1982). “On the number of Markoff numbers below a given bound”. Mathematics of Computation 39 (160): pp. 709–723 .
- ↑ K. Soundararajan (1993). “Primes in a sparse sequence”. Journal of Number Theory 43 (2): pp. 220–227.
- ↑ Morris Newman, Daniel Shanks, H. C. Williams, Simple groups of square order and an interesting sequence of primes, Acta Arithmetica 38 (1980/1981), pp. 129–140.
- ↑ Glyn Harman (2012). “Counting primes whose sum of digits is prime”. Journal of Integer Sequences 15: pp. 7 pp .
- ↑ William D. Banks; Derrick N. Hart; Mayumi Sakata (2004). “Almost all palindromes are composite”. Mathematical Research Letters 11: pp. 853–868 .
- ↑ Dmitry Mavlo (1995). “Absolute prime numbers”. The Mathematical Gazette 79 (485): pp. 299–304.
- ↑ Chris Caldwell, Repunit on the Prime Pages
- ↑ 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.
- ↑ Peter Fletcher; William Lindgren; Carl Pomerance (1996). “Symmetric and asymmetric primes”. Journal of Number Theory 58 (1): pp. 89–99.
- ↑ 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.
- ↑ 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.
- ↑ 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.
- ↑ 23.0 23.1 Matthew Just, On upper bounds for the count of elite primes, Integers 20 (2020).
- ↑ Tom Müller (2006). “Searching for large elite primes”. Experimental Mathematics 15 (2): pp. 183–186.
- ↑ 25.0 25.1 25.2 Kevin Broughan and Zhou Qizhi (2010). “Flat primes and thin primes”. Bulletin of the Australian Mathematical Society 82 (2): pp. 282–292 .
- ↑ Carl Pomerance (1979). “The prime number graph”. Mathematics of Computation 33 (145): pp. 399–408 .
- ↑ Ben Green; Terence Tao (2008). “The primes contain arbitrarily long arithmetic progressions”. Annals of Mathematics 167 (2): pp. 481–547. arXiv:math/0404188 .
- ↑ Ben Green and Terence Tao, A bound for progressions of length k in the primes
- ↑ Andrew Granville (2008). “Prime number patterns”. Amer. Math. Monthly 115 (4): pp. 279–296 .
- ↑ David W. Boyd (1994). “A p-adic study of the partial sums of the harmonic series”. Experimental Mathematics 3 (4): pp. 287–302 ..
- ↑ 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..
- ↑ K. L. Jensen (1915). “Om talteoretiske Egenskaber ved de Bernoulliske Tal”. Nyt Tidsskrift für Matematik Afdeling B 28: pp. 73–83.
- ↑ 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 .
- ↑ Adam P. Goucher, Ménage primes (2014)
- ↑ G. E. Hardy; M. V. Subbarao (2002). “A modified problem of Pillai and some related questions”. American Mathematical Monthly 109 (6): pp. 554–559.
- ↑ Srinivas Ramanujan (1919). “A proof of Bertrand's postulate”. Journal of the Indian Mathematical Society 11: pp. 181–182 .
- ↑ Jonathan Sondow (2009). “Ramanujan Primes and Bertrand's Postulate”. American Mathematical Monthly 116: pp. 630–635. arXiv:0907.5232.
- ↑ Kevin A. Broughan; A. Ross Barnett (2009). “On the subsequence of primes having prime subscripts”. Journal of Integer Sequences 12. article 09.2.3 .
- ↑ 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.
- ↑ R. W. K. Odoni (1985). “On the prime divisors of the sequence ”. Journal of the London Mathematical Society 32 (1): pp. 1–11.
- ↑ W. Banks, P. Pollack, & C. Pomerance, Symmetric primes revisited, Integers 19 (2019), A54, 7 pp.
- ↑ Richard J. McIntosh and Eric L. Roettger (2007). “A search for Fibonacci–Wieferich and Wolstenholme primes”. Mathematics of Computation 76: pp. 2087–2094.
- ↑ Jiří Klaška (2008). “Short remark on Fibonacci-Wieferich primes”. Acta Mathematica et Informatica Universitatis Ostraviensis 15 (15): pp. 21–25 .
- ↑ Grell, George; Peng, Wayne (2015). “Wall's Conjecture and the ABC Conjecture”. arΧiv:1511.01210.
- ↑ 45.0 45.1 R. Crandall; K. Dilcher; C. Pomerance (1997). “A search for Wieferich and Wilson primes”. Mathematics of Computation 66 (217): pp. 433–449 .
- ↑ Joshua Knauer and Jörg Richstein (2005). “The continuing search for Wieferich primes”. Mathematics of Computation 74: pp. 1559–1563 .
- ↑ R. J. McIntosh (1995). “On the converse of Wolstenholme's Theorem”. Acta Arithmetica 71: pp. 381–389 .
- ↑ Binbin Zhou (2009). “The Chen primes contain arbitrarily long arithmetic progressions”. Acta Arithmetica 138: pp. 301–315 .