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List of prime conjectures

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This page was created to organize all the conjectures and unsolved problems involving prime numbers, listed from highest to least importance. If new, relevant conjectures are made, they may be added to this page. If one of the problems is solved and accepted by the mathematics community, then they may be removed.

Name of problem Who first posed the problem Date proposed Notes
Riemann Hypothesis Bernhard Riemann 1859

[R] Bernhard Riemann, "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse", (1859) Monatsberichte der Berliner Akademie.
[T] E.C. Titchmarsh, "The theory of the Riemann zeta function", Oxford. Univ Press
[P] G. Polya, "Bemerkung ueber die Integraldarstellung der Riemannsche zeta-Funktion", Acta Math. 48 (1926), 305-317.
[B] D. Bump, K.K. Choi, P. Kurlberg, J. Vaaler, "A Local Riemann Hypothesis, Math. Zeit. 233 (2000) p1-19.
[H] H. Hamburger, "Ueber die Riemannsche Funktionalgleichung der zeta-Funktion", Math. Zeit. 10 (1921), 240-254.
[V] A. Karatsuba, Voronin S., "The Riemann Zeta function, De Gruyter Exposition of Mathematics, Transl. Neil Koeblitz (1975) p212.
[F] J. Faraut, A. Koranyi, "Function spaces and reproducing kernels on bounded symmetric domains, J. Funct. An. 88 (1990) p64-89.

Goldbach conjecture Christian Goldbach 1742 A weaker version of the conjecture was proven here
Twin prime conjecture Supposedly by Alphonse de Polignac 1849
Finitude of the Fermat primes 1859
Polignac's conjecture Alphonse de Polignac 1849 See [P] A. de Polignac, Six propositions arithmologiques déduites de crible d'Ératosthène. Nouv. Ann. Math. 8 (1849), pp. 423--429
Infinitude of -primes Edmund Landau 1912 A002496
Legendre's conjecture Adrien-Marie Legendre
Brocard's conjecture Henri Brocard A050216
Firoozbakht's conjecture Farideh Firoozbakht 1982 Rivera, Carlos. "Conjecture 30. The Firoozbakht Conjecture"

Oliveira e Silva, Tomás. Gaps between consecutive primes
Kourbatov, Alexei. Verification of the Firoozbakht conjecture for primes up to 4×1018 - arXiv:1503.01744
Kourbatov, Alexei. Upper bounds for prime gaps related to Firoozbakht's conjecture - arXiv:1506.03042

Andrica's conjecture Dorin Andrica 1986 Andrica, D. (1986). "Note on a conjecture in prime number theory". Studia Univ. Babes-Bolyai Math. 31 (4): 44–48.

Prime Numbers: The Most Mysterious Figures in Math, John Wiley & Sons, Inc., 2005, p. 13.

Cramér's conjecture Harald Cramér 1936 Cramér, Harald (1936), "On the order of magnitude of the difference between consecutive prime numbers"

Westzynthius, E. (1931), "Über die Verteilung der Zahlen die zu den n ersten Primzahlen teilerfremd sind", Commentationes Physico-Mathematicae Helingsfors 5: 1–37
Shanks, Daniel (1964), "On Maximal Gaps between Successive Primes", Mathematics of Computation (American Mathematical Society) 18 (88): 646–651
Granville, Andrew (1995), "Harald Cramér and the distribution of prime numbers"
Nicely, Thomas R. (1999), "New maximal prime gaps and first occurrences", Mathematics of Computation 68 (227): 1311–1315

Oppermann's conjecture Ludvig Oppermann 1882 Oppermann, L. (1882), "Om vor Kundskab om Primtallenes Mængde mellem givne Grændser", Oversigt over det Kongelige Danske Videnskabernes Selskabs Forhandlinger og dets Medlemmers Arbejder: 169–179
Counterexamples to the Baillie–PSW primality test Robert Baillie Carl Pomerance. "Are There Counterexamples to the Baillie-PSW Primality Test?"

Thomas R. Nicely. " The Baillie-PSW primality test."
R. K. Guy. "Pseudoprimes. Euler Pseudoprimes. Strong Pseudoprimes". §A12 in "Unsolved Problems in Number Theory", 2nd ed. New York: Springer-Verlag, pp. 27-30, 1994.

Agoh–Giuga conjecture Giuseppe Giuga, Takashi Agoh 1950 Borwein, D.; Borwein, J. M.; Borwein, P. B.; Girgensohn, R. (1996). "Giuga's Conjecture on Primality". American Mathematical Monthly 103: 40–50

Giuga, Giuseppe (1951). "Su una presumibile proprietà caratteristica dei numeri primi". Ist.Lombardo Sci. Lett., Rend., Cl. Sci. Mat. Natur. (in Italian) 83: 511–518

Infinitude of Wall–Sun–Sun primes Donald Dines Wall, Zhi Hong Sun, Zhi Wei Sun Andrejić, V. (2006). On Fibonacci powers. Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. 17: 38–44. doi:10.2298/PETF0617038A.

Wall–Sun–Sun Prime Search project at PrimeGrid
Wall-Sun-Sun Prime Search statistics at PrimeGrid

See also list of resolved prime conjectures.