Notation:

ℙ the set of the prime numbers in ℕ ℕ the set of the natural numbers, i.e. positive integers ℤ the set of the integers, i.e. the whole numbers ℚ the set of the rational numbers, i.e. fractional numbers ℝ the set of the reell numbers ℂ the set of the complex numbers

μ(n) the Möbius function μ: ℕ → {−1, 0, 1} Li(x) the Integral-logarithm ∫_{2}^{x} \frac{1}{ln t}dt π(n) the prime counting function ℤ[\sqrt{d}] for all square-free d ∈ ℤ the set \{i + j\sqrt{d} | i, j ∈ ℤ \}

k|i means k divides i, i.e. ∃ j ∈ ℤ : i = j·k ζ(z) The function ∑_{n=1}^{∞} \frac{1}{n^{z}} which is unique analytic extensible to all z ∈ ℂ\\{1\} |z| means the norm of the number z, e.g. the absolut value of z if z ∈ ℝ |S| means the cardinality of the set S

Important Mathematical Conjectures in the Past and the Present

-11. Euler’s sum-of-power conjecture

date: — disproved 1967 category: number theory formally: ∀ n ∈ ℕ : ∃ k, a ∈ ℕ ∃ a_{i} ∈ ℕ : ∑_{i=1}^{k} a_{i}^{n} = a^{n} ⇒ k ≥ n or k = 1

-10. Waring’s conjecture

date: 1789 — proved 1909 category: number theory statement: each sufficient large integer m is representable as the sum of at most k n-th

powers, where k dependens only on n. formally: ∀ n ∈ ℕ ∃ k ∈ ℕ ∀ m ∈ ℕ : m = ∑_{i=1}^{k} a_{i}^{n} with a_{i} ∈ ℕ

-9. Riemann’s conjecture

date: — disproved 1914 category: number theory formally: ∀ n ∈ ℕ : π(n) ∼ ∑_{i=1}^{∞} \frac{μ(i)}{i}Li(n^{\frac{1}{i}}) ⇒ ∀ n ∈ ℕ : π(n) < Li(n)

-8. Borsuk’s conjecture

date: 1911 — disproved 1993 category: geometry statement: Each n-dimensional convex set of diameter 1 can be covered by n+1 convex

sets such that each has diameter strictly < 1. -7. Burnside’s conjecture

category: group theory statement: Each non-commutative simple group has an even number of elements.

date: 1902 — proved 1963 -6. Mertens’ conjecture

date: 1897 — disproved 1985 category: number theory formally: ∀ n ∈ ℕ : | ∑_{i=1}^{n} μ(i)| ≤ \sqrt{n}

-5. Gauss’ class number conjecture

date: 1801 — proved 1934/1936 category: number theory statement: Each class-number has only finite many solutions for ℤ[\sqrt{−d}] formally: ∀ n ∈ ℕ ∀ D ⊂ ℕ ∀ d ∈ D : h(ℤ[\sqrt{−d}]) = n ⇒ |D| < ∞ -4. Bieberbach conjecture

date: 1916 — proved 1985 category: function theory statement: The n-th coefficient in the formal power series of a univalent function is

≤ n. -3. Mordell conjecture

date: 1922 — proved 1984 category: number theory statement: Each diophantic equation in k variables which root-space in ℂ^{k} is neither

simple nor double connected has only finite many non-trivial solutions in ℤ. -2. Fermat conjecture

date: 1637 — proved 1995 category: number theory formally: ∀ n, x, y, z ∈ ℕ : x^{n} + y^{n} = z^{n} ⇒ n = 1, 2

-1. Four-color conjecture

date: 1853 — proved 1976 category: graph theory statement: Each map which is embedable in the Euclidean plane is colorable by at most

four colors , such that no face has a common edge with an adjacent face. formally: Let G(V, E) be a plane graph. Then for G holds:

∃ f: V →C ∀ e=(v, w) ∈ E : f(v) ≠ f(w) and |C| ≤ 4

0. Continium hypothesis

date: 1877 — clarified 1931/1963 category: set theory formally: |∃ set S : |ℕ| < |S| < |2^{ℕ}| = |ℝ|

1. Riemann hypothesis

date: 1859 category: number theory / function theory statement: All non-trivial zeroes of the Riemann-Zetafunction have real-part one half. formally: x ∈ ℂ\ℝ and ζ(x) = 0 ⇒ x = 1/2 + it and t ∈ ℝ

2. Goldbach conjecture

date: 1742 category: number theory statement: Each even integer > 2 is representable by the sum of two primes. formally: ∀ n ∈ ℕ ∃ p, q ∈ ℙ : 2n = p + q or 2n = p

3. Odd perfect number conjecture

date: 1747(?) category: number theory statement: There is no odd perfect number formally: ∀ n ∈ ℕ : ∑_{d∈ℕ: d|n} d = 2n ⇒ 2 | n

4. Poincarè conjecture

date: 1904 category: topology statement: A simple-connected n-dimensional manifold is homeomorphic to the surface

of a n+1-dimensional ball for every n ∈ ℕ.

5. Kepler conjecture

date: 1611 category: geometry statement: The asymptotical densest sphere packing in ℝ^{3} is given by the combinations

of layers of hexagonal closed packed circles in ℝ^{2}.

6. Twin-Prime conjecture date: 1922 category: number theory statement: There are infinite many prime-twins formally: ∀ m ∈ ℕ ∃ n ∈ ℕ : p_{n} ∈ ℙ, p_{n} ≥ m and p_{n} + 2 ∈ ℙ

7. Taniyama-Shimura conjecture

date: 1955 category: number theory statement: For every elliptic curve, there is a modular form with the same Dirichlet

L-series as the elliptic curve.

8. Euler’s conjecture

date: 1912(?) category: number theory statement: if g(n) denotes the minimal k in Waring’s problem,

then ∀ n ∈ ℕ : g(n) = 2^{n} + ⌊1.5^{n}⌋ − 2

9. Catalan conjecture

date: 1844 category: number theory formally: ∀ a, b, n, m ∈ ℕ\{1} : a^{n} − b^{m} = 1 ⇒ a = m = 3, b = 2 = 1

10. Schinzel conjecture category: number theory formally: ∀ n ∈ ℕ : n ≤ 8 or p_{n+1} − p_{n} > ln^{2}(p_{n})

11. Mersenne prime conjecture

date: 19?? category: number theory statement: There exist infinitely many primes of the form 2^{p}−1 with p ∈ ℙ.

12. Birch-Swinnerton-Dyer conjecture

date: 1962 category: number theory

Only statements ranked which are at least 20 years old, are understood by at least hundred mathematicians and are/were believed to be true by a majority (at a certain time).

• abc Conjecture (open)

• Abhyankar’s Conjecture

• Ablowitz-Ramani-Segur Conjecture

• Andrica’s Conjecture (open)

• Annulus Conjecture

• Argoh’s Conjecture

• Artin’s Conjecture (open)

• Bachet’s Conjecture (proved)

• Bennequin’s Conjecture

• Bieberbach Conjecture (proved)

• Birch Conjecture (open)

• Blaschke Conjecture

• Borsuk’s Conjecture (disproved)

• Borwein Conjectures

• Braun’s Conjecture

• Brocard’s Conjecture

• Burnside’s Conjecture (proved)

• Carmichael’s Conjecture

• Catalan’s Conjecture (open)

• Cramér Conjecture (open)

• de Polignac’s Conjecture (open)

• Dodecahedral Conjecture (open)

• Double Bubble Conjecture

• Eberhart’s Conjecture

• Euler’s Conjecture (open)

• Euler Power Conjecture (disproved)

• Euler Quartic Conjecture (disproved)

• Feit-Thompson Conjecture

• Fermat’s Conjecture (proved)

• Flyping Conjecture

• Gilbreath’s Conjecture

• Giuga’s Conjecture

• Goldbach Conjecture (open)

• Grimm’s Conjecture

• Guy’s Conjecture

• Hardy-Littlewood Conjectures (open)

• Hasse’s Conjecture (open)

• Heawood Conjecture (proved)

• Jacobian Conjecture

• Kaplan-Yorke Conjecture

• Keller’s Conjecture

• Kelvin’s Conjecture

• Kepler Conjecture (open)

• Kreisel Conjecture

• Kummer’s Conjecture

• Local Density Conjecture

• Mertens’ Conjecture (disproved)

• Milin Conjecture

• Milnor’s Conjecture (proved)

• Mordell Conjecture (proved)

• Netto’s Conjecture

• Nirenberg’s Conjecture

• Ore’s Conjecture

• Padé Conjecture

• Palindromic Number Conjecture

• Pillai’s Conjecture

• Poincaré Conjecture (open)

• Pólya Conjecture (disproved)

• Prime k-Tuples Conjecture (open)

• Prime Power Conjecture

• Quillen-Lichtenbaum Conjecture

• Ramanujan-Petersson Conjecture • Robertson Conjecture

• Shafarevich Conjecture

• Sausage Conjecture (open)

• Schanuel’s Conjecture

• Scholz Conjecture

• Seifert Conjecture

• Selfridge’s Conjecture

• Shanks’ Conjecture

• Smith Conjecture

• Swinnerton-Dyer Conjecture (open)

• Szpiro’s Conjecture • Tait’s Hamiltonian Graph Conjecture (disproved)

• Tait’s Knot Conjectures

• Taniyama-Shimura Conjecture (open)

• Tau Conjecture (proved)

• Tau Dirichlet Series Conjecture (open)

• Thurston’s Geometrization Conjecture

• Thwaites’ Conjecture (open)

• Vojta’s Conjecture

• Wang’s Conjecture (disproved)

• Waring’s Prime Conjecture (open)

• Waring’s Sum Conjecture (proved)

• Zarankiewicz’s Conjecture

• Bourget’s Hypothesis (open)

• Continuum Hypothesis (the only “ever-to-be” hypothesis among the here listed)

• Ramanujan’s Hypothesis (proved)

• Riemann Hypothesis (open)

• Schinzel’s Hypothesis (open)

• Souslin’s Hypothesis For this list of conjecture names, I’m gratefully to Eric W. Weisstein. Their meanings is documented in his magnificant mathematical lexicon at the URL

http://www.astro.virginia.edu/∼eww6n/math/Conjecture.html.
Have a look at http://www.sci.net.mx/∼crivera/ppp/conjectures.htm, also.

1998-07-12 18:10 UT+2