Wiles numbers are easy to bound, yet non-trivial to compute.

Rules:

  • The Wiles number of Wiles is 0.
  • Your Wiles number is 1 + the smallest Wiles number of your co-authors.

    A popular route is via Paul Erdos:

  • The Wiles number of Erdos is not greater than 3.
  • My Erdos number is not greater than 3.
  • Thus my Wiles number is not greater than 6.

    For example, a 6-link route is provided by:

    0) A.J.Wiles, Modular elliptic curves and Fermat's last theorem, Ann.Math. 141 (1995) 443-551.

    1) C.M.SKINNER, A.J.Wiles, Ordinary representations and modular forms, Proc.Nat.Acad.Sci. 94 (1997) 10520-10527.

    2) A.M.ODLYZKO, C.M. Skinner, Nonexistence of Siegel zeros in towers of radical extensions, Contemp.Math. 143 (1993) 499-511.

    3) P.ERDOS, A.Hildebrand, A.Odlyzko, P.Pudaite, B.Reznick, The asymptotic behavior of a family of sequences, Pacific J.Math. 126 (1987) 227-241.

    4) H.G.DIAMOND, P.Erdos, On sharp elementary prime number estimates, Enseign.Math. 26 (1980) 313-321.

    5) D.M.BRADLEY, H.G.Diamond, A difference differential equation of Euler-Cauchy type, J.Diff.Equations 138 (1997) 267-300.

    6) J.M.Borwein, D.M.Bradley, D.J.Broadhurst, Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k, Electron.J.Combin. 4 (1997) R5.

    However, Maths Abstracts shows that my Wiles number is not greater than 5.

    A 5-link route is provided by:

    1) B.MAZUR, A.J.Wiles, On p-adic analytic families of Galois representations, Compositio Math. 59 (1986) 231-264.

    2) A.M.Gleason, A.JAFFE, B.Mazur, R.H.Herman, C.H.Clemens, J.Kollar, K.Gawedzki, C.Soule, M.Sipser, ICM-90, Report on the International Congress of Mathematicians held in Kyoto, August 21-29, 1990, Notices Amer.Math.Soc. 37 (1990) 1209-1216.

    3) J.Lebowitz, M.Atiyah, E.Brezin, A.CONNES, J.Froehlich, D.Gross, A.Jaffe, L.Kadanoff, D.Ruelle, Round table: physics and mathematics, XIth International Congress of Mathematical Physics (Paris, 1994), Internat. Press, Cambridge, MA, 1995, 691-705.

    4) A.Connes, D.KREIMER, Hopf algebras, renormalization and noncommutative geometry, Comm.Math.Phys. in press.

    5) D.J.Broadhurst, D.Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Phys.Lett. B393 (1997) 403-412.

    David Broadhurst

    14 March 1999