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	http://www.claymath.org/millennium/Riemann_Hypothesis/ [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 $n^{2}+1$-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 http://arxiv.org/abs/1102.1636

See also list of resolved prime conjectures.