User:Charles R Greathouse IV/Tables of special primes

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 $x^{3}+k$ 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 ${\tfrac {1}{4}}(3n^{2}+1)$	conjectured infinite with density $C{\sqrt {n}}/\log n$ : Hardy-Littlewood conjecture F	A002407
Landau primes $n^{2}+1$	conjectured infinite with density $C{\sqrt {n}}/\log n$ (C ≈ 1.372813): Hardy-Littlewood conjecture E	A002496
Central polygonal primes $n^{2}-n+1$	conjectured infinite with density $C{\sqrt {n}}/\log n$ : Hardy-Littlewood conjecture F	A002383
Centered triangular primes ${\tfrac {1}{2}}(3n^{2}+3n+2)$	conjectured infinite with density $C{\sqrt {n}}/\log n$ : Hardy-Littlewood conjecture F	A125602
Centered square primes ${\tfrac {1}{2}}(4n^{2}+4n+2)$	conjectured infinite with density $C{\sqrt {n}}/\log n$ : Hardy-Littlewood conjecture F	A027862
Centered pentagonal primes ${\tfrac {1}{2}}(5n^{2}+5n+2)$	conjectured infinite with density $C{\sqrt {n}}/\log n$ : Hardy-Littlewood conjecture F	A145838
Centered hexagonal primes ${\tfrac {1}{2}}(6n^{2}+6n+2)$	conjectured infinite with density $C{\sqrt {n}}/\log n$ : Hardy-Littlewood conjecture F	A002407
Cuban primes (variant) $3n^{2}+1$	conjectured infinite with density $C{\sqrt {n}}/\log n$ : Hardy-Littlewood conjecture F	A002648
Centered heptagonal primes ${\tfrac {1}{2}}(7n^{2}+7n+2)$	conjectured infinite with density $C{\sqrt {n}}/\log n$ : Hardy-Littlewood conjecture F	A144974
Centered decagonal primes ${\tfrac {1}{2}}(10n^{2}+10n+2)$	conjectured infinite with density $C{\sqrt {n}}/\log n$ : Hardy-Littlewood conjecture F	A090562
Star primes $6n^{2}-6n+1$	conjectured infinite with density $C{\sqrt {n}}/\log n$ : Hardy-Littlewood conjecture F	A083577
Cubic	Size	A-number
$n^{3}+2$	conjectured infinite with density $C{\sqrt[{3}]{n}}/\log n$ (C ≈ 1.2985): Hardy-Littlewood conjecture K	A144953
Exponential	Size	A-number
Wagstaff primes ${\tfrac {1}{3}}(2^{n}+1)$	unknown, trivial density O(log n)	A000979
Mersenne primes $2^{n}-1\,$	conjectured infinite with density log₂ log n + o(log log n): Lenstra–Pomerance–Wagstaff conjecture	A000668
Thābit primes $3\cdot 2^{n}-1$	unknown, trivial density O(log n)	A007505
Cullen primes $n\cdot 2^{n}+1$	unknown, density o(log n)^[1]	A050920
Woodall primes $n\cdot 2^{n}-1$	unknown, density o(log n)^[2]	A050918
Super-exponential	Size	A-number
Double Mersenne primes ${\tfrac {1}{2}}\cdot 2^{2^{n}}-1$	unknown, trivial density O(log log n/log log log n)
Legendre primes $\lfloor A^{2^{n}}\rfloor$ for $A=1.524\ldots$	infinite with density Θ(log log n) under Legendre's conjecture; otherwise possibly ill-defined	A059784
Fermat primes $2^{2^{n}}+1$	conjectured finite (5 known)	A019434
Mills primes $\lfloor A^{3^{n}}\rfloor$ for $A=1.306\ldots$	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 $m^{2}+mn+n^{2}$	-3	infinite with density 0.5n/log n + O(n/log² n): Dirichlet's theorem	A007645	Positive definite
$m^{2}+3n^{2}$	-12		A007645	Positive definite
$m^{2}+n^{2}$	-4	0.5n/log n + O(n/(log n)²): Fermat's theorem on sums of two squares	A002313	Positive definite
$m^{2}+4n^{2}$ (excluding 2)	-16		A002313	Positive definite
$m^{2}+mn+2n^{2}$	-7		A106856	Positive definite
$2m^{2}+2mn+17n^{2}$	-132	infinite with density 0.15n/log n + O(n/log² n): Dirichlet's theorem	A139827	Positive definite
$m^{2}+mn-2n^{2}$	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 $m^{2}+n^{4}$	Θ(n^3/4/log n)^[4]	A028916
Half-octavan primes $(m^{8}+n^{8})/2$	trivial density O(n^1/4), conjectured infinite	A290780
Half-quartan primes $(m^{4}+n^{4})/2$	trivial density O(n^1/2), conjectured infinite	A002646
Heath-Brown primes $m^{3}+2n^{3}$	Θ(n^2/3/log n)^[5]	A173587
Leyland primes $m^{n}+n^{m}\,$ (m, n > 1)	trivial density O((log n)²log log n)	A094133
Octavan primes $m^{8}+n^{8}$	trivial density O(n^1/4), conjectured infinite	A006686
Pierpont primes $2^{m}3^{n}+1$	conjectured infinite^[6] with density Θ(log n), trivial density O(log² n)	A005109
Proth primes $n\cdot 2^{m}+1$ with $n<2^{m}$	trivial density O(n^1/2)	A080076
Quartan primes $m^{4}+n^{4}$	trivial density O(n^1/2), conjectured infinite	A002645
Semi-octavan primes $m^{4}+n^{8}$	trivial density O(n^3/8), conjectured infinite	A291206
Solinas primes $2^{m}\pm 2^{n}\pm 1$ with $0<n<m$	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 $P=n!-(n-1)!+\cdots \pm 1!$	finite^[7]	A071828
Bertrand primes $\lfloor 2^{b}\rfloor ,\lfloor 2^{2^{b}}\rfloor ,\ldots$ for $b=1.251\ldots$	infinite by Bertrand's postulate with density lg ⁎ n + O(1)	A051501
Euclid primes $P=n\#+1$	conjectured infinite with density e^γ log n^[8]	A018239
Factorial primes $P=n!-1\,$	conjectured infinite with density e^γ log n^[8]	A055490
Factorial primes $P=n!+1\,$	conjectured infinite with density e^γ log n^[8]	A088054
Fouvry–Iwaniec primes $P=x^{2}+p^{2}$	infinite with density Θ(n/(log n)²)^[9]	A185086
Heath-Brown–Li primes $P=n^{2}+p^{4}\,$	Θ(n^3/4/log² n)^[10]	A281792
Markov primes $m^{2}+n^{2}+P^{2}=3mnP$	trivial density O((log n)²)^[11]	A178444
Primorial primes $P=n\#-1$	conjectured infinite with density e^γ log n^[8]	A057705
Primorial primes $P=n\#+1$	conjectured infinite with density e^γ log n^[8]	A005234
Soundararajan primes $P=1^{1}+2^{2}+\cdots +n^{n}$	density O(log n/(log log n)²)^[12]	A073826
Three-square primes $P=l^{2}+m^{2}+n^{2}$	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 $a_{n}=a_{n-1}+a_{n-2}$ with $a_{1}=a_{2}=1$	density O(log n/log log n)	A005478
Lucas primes $a_{n}=a_{n-1}+a_{n-2}$ with $a_{1}=1,a_{2}=3$	density O(log n/log log n)	A005479
Padovan primes $a_{n}=a_{n-2}+a_{n-3}$ with $a_{0}=a_{1}=a_{2}=1$	trivial density O(log n)	A100891
NSW primes $a_{n}=2a_{n-1}+a_{n-2}$ with $a_{0}=a_{1}=1$	density O(log n/log log n)^[13]	A088165
Pell primes $a_{n}=2a_{n-1}+a_{n-2}$ with $a_{1}=1,a_{2}=2$	density O(log n/log log n)	A086383
Perrin primes $a_{n}=a_{n-2}+a_{n-3}$ with $a_{1}=0,a_{1}=2,a_{2}=3$	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 $cn/(\log n)^{k}$ , where c is an effectively computable constant depending only on the form of the constellation.

	Size	A-number
Twin primes $(p,p+2)$	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 $(p,p+4)$	conjectured infinite with density 2C₂n/(log n)²	A023200
Sexy primes $(p,p+6)$	conjectured infinite with density 4C₂n/(log n)²	A023201
Prime triplets $(p,p+2,p+6)$	conjectured infinite with density 4.5C₃n/(log n)³	A022004
Prime triplets $(p,p+4,p+6)$	conjectured infinite with density 4.5C₃n/(log n)³	A022005
Prime quadruplets $(p,p+2,p+6,p+8)$	conjectured infinite with density 13.5C₄n/(log n)⁴	A007530

Primes by size

	Size	A-number
Odd primes $p>2$	infinite with density n/log n + O(n/(log n)²): prime number theorem	A065091
Titanic primes $p>10^{999}$	infinite with density n/log n + O(n/(log n)²): prime number theorem	A074282+10⁹⁹⁹
Gigantic primes $p>10^{9999}$	infinite with density n/log n + O(n/(log n)²): prime number theorem	A142587+10⁹⁹⁹⁹
Megaprimes $p>10^{999999}$	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^[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 C_Artinn/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(n^1/2log 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(n^0.955).	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-C_Artin)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/log^1.086 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(n^5/6) 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. http://eprints.maths.ox.ac.uk/168/.
↑ 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 $\sum _{i=1}^{n}(-1)^{n-i}i!$ 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 $a^{2}+p^{4}$ ”. Inventiones Mathematicae 208 (2): pp. 1–59. https://arxiv.org/abs/1504.00531.
↑ 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 $\scriptstyle w_{n+1}=1+w_{1}...w_{n}$ ”. 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.

[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] 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.

[5] D. R. Heath-Brown (2001). “Primes represented by x³ + 2y³”. Acta Mathematica 186: pp. 1–84. http://eprints.maths.ox.ac.uk/168/.

[6] Andrew M. Gleason (1988). “Angle trisection, the heptagon, and the triskaidecagon”. The American Mathematical Monthly 95 (3): pp. 185–194.

[7] Miodrag Živković (1999). “The number of primes $\sum _{i=1}^{n}(-1)^{n-i}i!$ is finite”. Mathematics of Computation 68: pp. 403–409.

[CG2002-8] 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.

[9] Étienne Fouvry and Henryk Iwaniec (1997). “Gaussian primes”. Acta Arithmetica 79 (3): pp. 249–287. .

[10] D. R. Heath-Brown; Xiannan Li (2017). “Prime values of $a^{2}+p^{4}$ ”. Inventiones Mathematicae 208 (2): pp. 1–59. https://arxiv.org/abs/1504.00531.

[11] Don Zagier (1982). “On the number of Markoff numbers below a given bound”. Mathematics of Computation 39 (160): pp. 709–723.

[12] K. Soundararajan (1993). “Primes in a sparse sequence”. Journal of Number Theory 43 (2): pp. 220–227.

[13] 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.

[14] Glyn Harman (2012). “Counting primes whose sum of digits is prime”. Journal of Integer Sequences 15: pp. 7 pp.

[15] William D. Banks; Derrick N. Hart; Mayumi Sakata (2004). “Almost all palindromes are composite”. Mathematical Research Letters 11: pp. 853–868.

[16] Dmitry Mavlo (1995). “Absolute prime numbers”. The Mathematical Gazette 79 (485): pp. 299–304.

[17] Chris Caldwell, Repunit on the Prime Pages

[18] 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.

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User:Charles R Greathouse IV/Tables of special primes

Contents

Diophantine primes

Simple univariate Diophantine primes

Primes in quadratic bivariate forms

Simple bivariate Diophantine primes

Other Diophantine primes

Recurrence relation primes

Prime constellations

Primes by size

Base-dependent primes

Other classes of primes

References

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