Number Theory — Open Conjectures

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# Conjecture Eponym(s) Comments
1 abc conjecture David Masser, Joseph Oesterlé ⇔ Granville–Langevin conjecture, Vojta’s conjecture in dimension 1; ⇒ Erdős–Woods conjecture, Fermat–Catalan conjecture. Proof claimed in 2012 by Shinichi Mochizuki.
2 Agoh–Giuga conjecture Takashi Agoh, Giuseppe Giuga  
3 Agrawal’s conjecture Manindra Agrawal  
4 Andrica’s conjecture Dorin Andrica  
5 Artin conjecture (L-functions) Emil Artin  
6 Artin’s conjecture on primitive roots Emil Artin ⇐ generalized Riemann hypothesis; ⇐ Selberg conjecture B
7 Bateman–Horn conjecture Paul T. Bateman, Roger Horn  
8 Beal’s conjecture Andrew Beal  
9 Beilinson conjecture Alexander Beilinson  
10 Birch and Swinnerton-Dyer conjecture Bryan John Birch, Peter Swinnerton-Dyer  
11 Birch–Tate conjecture Bryan John Birch, John Tate  
12 Bloch–Beilinson conjectures Spencer Bloch, Alexander Beilinson  
13 Brocard’s conjecture Henri Brocard  
14 Brumer–Stark conjecture Armand Brumer, Harold Stark  
15 Bunyakovsky conjecture Viktor Bunyakovsky  
16 Carmichael totient conjecture Robert Daniel Carmichael  
17 Catalan–Dickson conjecture on aliquot sequences Eugène Charles Catalan, Leonard Eugene Dickson  
18 Catalan’s Mersenne conjecture Eugène Charles Catalan  
19 Collatz conjecture Lothar Collatz  
20 Cramér’s conjecture Harald Cramér  
21 De Polignac’s conjecture Alphonse de Polignac  
22 Elliott–Halberstam conjecture Peter D. T. A. Elliott, Heini Halberstam  
23 Erdős–Straus conjecture Paul Erdős, Ernst G. Straus  
24 Firoozbakht’s conjecture Farideh Firoozbakht  
25 Fortune’s conjecture Reo Fortune  
26 Four exponentials conjecture —  
27 Gauss circle problem Carl Friedrich Gauss  
28 Gilbreath conjecture Norman Laurence Gilbreath  
29 Goldbach’s conjecture Christian Goldbach ⇒ The ternary Goldbach conjecture, which was the original formulation.
30 Goormaghtigh conjecture René Goormaghtigh  
31 Grimm’s conjecture Carl Albert Grimm  
32 Keating–Snaith conjecture Jonathan Keating, Nina Snaith  
33 Legendre’s conjecture Adrien-Marie Legendre  
34 Lemoine’s conjecture Émile Lemoine  
35 Lenstra–Pomerance–Wagstaff conjecture Hendrik Lenstra, Carl Pomerance, Samuel S. Wagstaff Jr.  
36 Leopoldt’s conjecture Heinrich-Wolfgang Leopoldt  
37 Marshall Hall’s conjecture Marshall Hall, Jr.  
38 Montgomery’s pair correlation conjecture Hugh Lowell Montgomery  
39 n conjecture —  
40 New Mersenne conjecture Marin Mersenne  
41 Oppermann’s conjecture Ludvig Oppermann  
42 Pillai’s conjecture Subbayya Sivasankaranarayana Pillai  
43 Riemann hypothesis Bernhard Riemann ⇐ Generalized Riemann hypothesis ⇐ Grand Riemann hypothesis; ⇔ De Bruijn–Newman constant = 0; ⇒ density hypothesis, Lindelöf hypothesis. See Hilbert–Pólya conjecture.
44 Sato–Tate conjecture Mikio Sato, John Tate  
45 Schanuel’s conjecture Stephen Schanuel  
46 Schinzel’s hypothesis H Andrzej Schinzel  
47 Second Hardy–Littlewood conjecture G. H. Hardy, John Edensor Littlewood  
48 Selfridge’s conjecture John Selfridge  
49 Twin prime conjecture —  
50 Unicity conjecture for Markov numbers Andrey Markov  
51 Vandiver’s conjecture Ernst Kummer, Harry Vandiver  
52 Vojta’s conjecture Paul Vojta ⇒ abc conjecture
53 Waring’s conjecture Edward Waring