User:CRGreathouse/Tables of special primes

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1 Simple Diophantine primes
2 Other Diophantine primes
3 Recurrence relation primes
4 Prime constellations
5 Primes by size
6 Base-dependent primes
7 Other classes of primes
8 References

[edit] Simple Diophantine primes

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

The Bunyakovsky conjecture, Schinzel's hypothesis H, and the fifth and sixth Hardy-Littlewood conjectures relate to the infinitude of the primes generated by certain polynomials.

Linear	Size	OEIS
Primes $n$	infinite with density $n / log n$ : prime number theorem	A000040
Real Eisenstein primes $3 n + 2$	infinite with density ½ in the primes: Dirichlet's theorem	A003627
Pythagorean primes $4 n + 1$	infinite with density ½ in the primes: Dirichlet's theorem	A002144
Real Gaussian primes $4 n + 3$	infinite with density ½ in the primes: Dirichlet's theorem	A002145
Quadratic	Size	OEIS
Landau primes $n 2 + 1$	conjectured infinite with density $1.3728\ldots\sqrt n/\log n$ : fifth Hardy-Littlewood conjecture	A002496
Hogben's central polygonal primes $n 2 - n + 1$	conjectured infinite with density $C\sqrt n/\log n$ : sixth Hardy-Littlewood conjecture	A002383
Centered triangular primes $\tfrac12(3n^2+3n+2)$	conjectured infinite?: Schinzel's hypothesis H	A125602
Centered square primes $\tfrac12(4n^2+4n+2)$	conjectured infinite with density $C\sqrt n/\log n$ : sixth Hardy-Littlewood conjecture	A027862
Centered pentagonal primes $\tfrac12(5n^2+5n+2)$	conjectured infinite?: Schinzel's hypothesis H
Star primes $6 n - 6 n + 1$	conjectured infinite with density $C\sqrt n/\log n$ : sixth Hardy-Littlewood conjecture	A083577
Centered hexagonal primes $\tfrac12(6n^2+6n+2)$	conjectured infinite with density $C\sqrt n/\log n$ : sixth Hardy-Littlewood conjecture	A002407
Cuban primes $3n^2+6n+4\,$	conjectured infinite with density $C\sqrt n/\log n$ : sixth Hardy-Littlewood conjecture	A002648
Centered heptagonal primes $\tfrac12(7n^2+7n+2)$	conjectured infinite?: Schinzel's hypothesis H
Centered decagonal primes $\tfrac12(10n^2+10n+2)$	conjectured infinite with density $C\sqrt n/\log n$ : sixth Hardy-Littlewood conjecture	A090562
Exponential	Size	OEIS
Wagstaff primes $\tfrac13(2^n+1)$	unknown, trivial density $\mathcal{O}(\log n)$	A000979
Mersenne primes $2^n-1\,$	conjectured infinite with density $\log_2(e^\gamma\cdot\log_2 n)$ : Lenstra–Pomerance–Wagstaff conjecture	A000668
Thābit primes $3\cdot2^n-1$	unknown, trivial density $\mathcal{O}(\log n)$	A007505
Cullen primes $n\cdot2^n+1$	unknown, trivial density $\mathcal{O}(\log n)$	A050920
Woodall primes $n\cdot2^n-1$	unknown, trivial density $\mathcal{O}(\log n)$	A050918
Carol primes $(2^n-1)^2-2\,$	unknown, trivial density $\mathcal{O}(\log n)$	A091516
Kynea primes $(2^n + 1)^2-2\,$	unknown, trivial density $\mathcal{O}(\log n)$	A091514
Super-exponential	Size	OEIS
Double Mersenne primes $\tfrac12\cdot2^{2^n}-1$	unknown, trivial density $\mathcal{O}(\log\log n)$
Fermat primes $2^{2^n}+1$	conjectured finite (5 known)	A019434
Mills primes $A^{3^n}$ for $A=1.306\ldots$	infinite by construction^[1] with density $Θ(loglog n)$	A051254

[edit] Other Diophantine primes

These are the prime solutions to more complicated Diophantine equations.

	Size	OEIS
Alternating factorial prime	finite (11 to 3,612,703 elements)^[2]	A071828
Primorial primes $P=n\#-1$	conjectured infinite with density $e γ log n$ ^[3]	A057705
Euclid primes $P=n\#+1$	conjectured infinite with density $e γ log n$ ^[3]	A018239
Factorial primes $P=n!-1\,$	conjectured infinite with density $e γ log n$ ^[3]	A055490
Factorial primes $P=n!+1\,$	conjectured infinite with density $e γ log n$ ^[3]	A088054
Leyland primes $P=m^n+n^m\,$ (m, n > 1)	unknown	A094133
Markov primes $m 2 + n 2 + P 2 = 3 m n p$	unknown
Pierpont primes $P = 2 m 3 n + 1$	conjectured infinite^[4] with density $Θ(log n)$	A005109
Proth primes $P=n\cdot2^m+1$ with $n < 2 m$	unknown	A080076
Soundararajan primes $P=1^1+2^2+\cdots+n^n$	density $\mathcal{O}(\log n/(\log\log n)^2)$ ^[5]	A073826

[edit] Recurrence relation primes

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

	Size	OEIS
Fibonacci primes $a n = a n - 1 + a n - 2$ with $a 1 = a 2 = 1$	density $o (log n)$ because only $F p$ can be prime; cardinality is an open question	A005478
Lucas primes $a n = a n - 1 + a n - 2$ with $a 1 = 1, a 2 = 3$	density $o (log n)$ because only $L p$ can be prime	A005479
Padovan primes $a n = a n - 2 + a n - 3$ with $a 0 = a 1 = a 2 = 1$	unknown	A100891
NSW primes $a n = 2 a n - 1 + a n - 2$ with $a 0 = a 1 = 1$	unknown	A088165
Pell primes $a n = 2 a n - 1 + a n - 2$ with $a 1 = 1, a 2 = 2$	unknown	A086383
Perrin primes $a n = a n - 2 + a n - 3$ with $a 1 = 0, a 1 = 2, a 2 = 3$	unknown	A074788

[edit] Prime constellations

The first Hardy-Littlewood conjecture states that each admissible prime k-tuple has an infinite number of primes of density $c n / (log n) k$ .

	Size	OEIS
Twin primes $(p, p + 2)$	density $\mathcal{O}\left(n\left(\frac{\log\log n}{\log n}\right)^2\right)$ : 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 quadruples $p, p + 2, p + 6, p + 8$	conjectured infinite with density 13.5C₄n/(log n)⁴	A007530

[edit] Primes by size

	Size	OEIS
Odd primes $p > 2$	infinite with density $n / log n$ : prime number theorem	A065091
Titanic primes $p > 10 999$	infinite with density $n / log n$ : prime number theorem	A074282+10⁹⁹⁹
Gigantic primes $p > 10 9999$	infinite with density $n / log n$ : prime number theorem
Megaprimes $p > 10 999999$	infinite with density $n / log n$ : prime number theorem

[edit] Base-dependent primes

	Size (base-10)	OEIS
Dihedral primes	density $\mathcal{O}((n/\log n)^{0.699})$ by the normality of the Copeland–Erdős constant	A038136
Emirps	conjectured density $Θ(n / (log n) 2)$ ^{[citation needed]}	A006567
Friedman primes	unknown	A112419
Full reptend primes	conjectured infinite with density $C_{\mathrm{Artin}}\approx0.374$ in the primes: Artin's conjecture on primitive roots	A001913
Left-truncatable primes	finite (4260 elements)	A024785
Happy primes		A035497
Minimal primes	finite (26 elements)	A071062
Palindromic primes	trivial density $\mathcal{O}(\sqrt n)$ , density 0 in the set of palindromes, cardinality is an open question^[6]	A002385
Pandigital primes	infinite with density 1 in the primes	A050288
Permutable primes	conjectured to be the repunit primes, plus a finite number of other primes	A003459
Primeval primes	unknown	A119535
Repunit primes	conjectured infinite with density $Θ(loglog n)$ ,^[7] trivial density $\mathcal{O}(\log n/\log\log n)$ since the length must be prime	A004022
Right-truncatable primes	finite (83 elements)	A024770
Self primes	unknown	A006378
Smarandache-Wellin prime	density $\mathcal{O}((n/\log n)^{0.699})$ by the normality of the Copeland–Erdős constant	A019518
Strobogrammatic primes	unknown, trivial density $\mathcal{O}(\log n)$	A007597
Unique primes	conjectured infinite	A040017
Weakly prime numbers	infinite with positive density in the primes^[8]	A050249

[edit] Other classes of primes

	Size	OEIS
Balanced primes	conjectured infinite	A006562
Bell primes	unknown	A051131
Chen primes	infinite^[9]	A109611
Elite primes	density $\mathcal{O}\left(\frac{n}{(\log n)^2}\right)$ ,^[10] conjectured infinite with density $\mathcal{O}(\log^c n)$ for $c\ge1$ ^[11]	A102742
Fortunate primes	conjectured infinite	A046066
Higgs primes	unknown	A007459
Highly cototient primes	unknown	A105440
Irregular primes	infinite; conjectured density $1 - e - 1 / 2$ in the primes	A000928
Lucky primes	unknown	A031157
Motzkin primes	unknown	A092832
Partition primes	unknown	A049575
Pillai primes	infinite^{[citation needed]}	A063980
Ramanujan primes	infinite^[12]	A104272
Regular primes	conjectured infinite with density $e - 1 / 2$ in the primes	A007703
Safe primes	conjectured infinite with density C₂/2(log n/2)²	A005385
Sophie Germain primes	conjectured infinite with density 2C₂/(log n)²	A005384
Stern primes	conjectured finite (8 known)	A042978
Super-primes	infinite, density $n / (log n) 2$	A006450
Supersingular primes	finite (15 elements)	A002267
Symmetric primes	density $\mathcal{O}(n/(\log n)^{1.027})$ ^[13], conjectured infinite	A090190
Ulam primes	believed to have positive density in the primes (see OEIS link)	A068820
Wedderburn-Etherington prime	trivial density $\mathcal{O}(\log n)$	A136402
Wieferich primes	heuristic density $Θ(loglog n)$ ^[14]	A001220
Wilson primes	heuristic density $Θ(loglog n)$ ^[14]	A007540
Wolstenholme primes	conjectured infinite^{[citation needed]}	A088164
Wall-Sun-Sun primes	heuristically infinite,^[15] none known

[edit] References

^ W. H. Mills, "A Prime-Representing Function." Bulletin of the American Mathematical Society 53 (1947), p. 604.
^ Miodrag Živković, "The number of primes $\sum_{i=1}^n(-1)^{n-i}i!$ is finite", Mathematics of Computation 68 (1999), pp. 403–409.
^ ^a ^b ^c ^d Chris K. Caldwell and Yves Gallot, "On the Primality of $n!\pm1$ and $2\times3\times5\times\cdots\times p\pm1$ ", Mathematics of Computation 71:237 (2002), pp. 441–448.
^ Andrew M. Gleason, "Angle Trisection, the Heptagon, and the Triskaidecagon". The American Mathematical Monthly 95:3 (1988), pp. 185–194. The conjecture is in a footnote on p. 191.
^ K. Soundararajan, "Primes in a Sparse Sequence", Journal of Number Theory 43:2 (1993), pp. 220–227.
^ William D. Banks, Derrick N. Hart, and Mayumi Sakata. "Almost All Palindromes Are Composite". Mathematical Research Letters 11 (2004), pp. 853–868.
^ Chris Caldwell, Repunit on the Prime Pages
^ Terence Tao, A remark on primality testing and decimal expansions, submitted to the Journal of the Australian Mathematical Society.
^ J. R. Chen, "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 (1973), pp. 157–176.
^ M. Křížek, F. Luca, and L. Somer. "On the convergence of series of reciprocals of primes related to the Fermat numbers", Journal of Number Theory 97 (2002), pp. 95–112.
^ Tom Müller, "Searching for large elite primes", Experimental Mathematics 15:2 (2006), pp. 183–186.
^ Srinivas Ramanujan, "A proof of Bertrand's postulate". Journal of the Indian Mathematical Society 11 (1919), pp. 181–182.
^ Peter Fletcher, William Lindgren and Carl Pomerance, "Symmetric and asymmetric primes", Journal of Number Theory 58:1 (1996), pp. 89–99.
^ ^a ^b R. Crandall, K. Dilcher and C. Pomerance, "A search for Wieferich and Wilson primes", Mathematics of Computation 66:217 (1997), pp. 433–449.
^ Chris Caldwell, Wall-Sun-Sun prime on the Prime Pages.