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Goldbach conjecture


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1: 27.13 Functions
§27.13(ii) Goldbach Conjecture
Goldbach’s assertion is that S ( n ) 1 for all even n > 4 . This conjecture dates back to 1742 and was undecided in 2009, although it has been confirmed numerically up to very large numbers. … The current status of Goldbach’s conjecture is described in the Wikipedia. … Hardy and Littlewood (1925) conjectures that G ( k ) < 2 k + 1 when k is not a power of 2, and that G ( k ) 4 k when k is a power of 2, but the most that is known (in 2009) is G ( k ) < c k ln k for some constant c . …
2: 18.38 Mathematical Applications
was used in de Branges’ proof of the long-standing Bieberbach conjecture concerning univalent functions on the unit disk in the complex plane. …
3: 16.23 Mathematical Applications
§16.23(iii) Conformal Mapping
The Bieberbach conjecture states that if n = 0 a n z n is a conformal map of the unit disk to any complex domain, then | a n | n | a 1 | . In the proof of this conjecture de Branges (1985) uses the inequality …
4: David M. Bressoud
 227, in 1980, Factorization and Primality Testing, published by Springer-Verlag in 1989, Second Year Calculus from Celestial Mechanics to Special Relativity, published by Springer-Verlag in 1992, A Radical Approach to Real Analysis, published by the Mathematical Association of America in 1994, with a second edition in 2007, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, published by the Mathematical Association of America and Cambridge University Press in 1999, A Course in Computational Number Theory (with S. …
5: Bibliography Y
  • A. J. Yee (2004) Partitions with difference conditions and Alder’s conjecture. Proc. Natl. Acad. Sci. USA 101 (47), pp. 16417–16418.
  • 6: 21.9 Integrable Equations
    Following the work of Krichever (1976), Novikov conjectured that the Riemann theta function in (21.9.4) gives rise to a solution of the KP equation (21.9.3) if, and only if, the theta function originates from a Riemann surface; see Dubrovin (1981, §IV.4). The first part of this conjecture was established in Krichever (1976); the second part was proved in Shiota (1986). …
    7: 17.14 Constant Term Identities
    Zeilberger–Bressoud Theorem (Andrews’ q -Dyson Conjecture)
    Macdonald (1982) includes extensive conjectures on generalizations of (17.14.1) to root systems. These conjectures were proved in Cherednik (1995), Habsieger (1986), and Kadell (1994); see also Macdonald (1998). …
    8: Richard A. Askey
    This inequality was a key element of Louis de Branges’ proof of the Bieberbach conjecture in 1985. …
    9: 17.13 Integrals
    Askey (1980) conjectured extensions of the foregoing integrals that are closely related to Macdonald (1982). These conjectures are proved independently in Habsieger (1988) and Kadell (1988).
    10: Bibliography K
  • K. W. J. Kadell (1988) A proof of Askey’s conjectured q -analogue of Selberg’s integral and a conjecture of Morris. SIAM J. Math. Anal. 19 (4), pp. 969–986.
  • K. W. J. Kadell (1994) A proof of the q -Macdonald-Morris conjecture for B C n . Mem. Amer. Math. Soc. 108 (516), pp. vi+80.
  • N. D. Kazarinoff (1988) Special functions and the Bieberbach conjecture. Amer. Math. Monthly 95 (8), pp. 689–696.