Abstract
For odd k, define θ(k) as the least value of s such that
has a non-trivial Solution over the integers. Fermat’s Last Theorem impl ies that θ(k) > 3 for odd k > 3.
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References
N.C. Ankeny, E. Artin, and S. Chowla. “The Class-Number of Real Quadratic Number Fields,” Ann. of Math. 56 (1952), 479–493.
This has the (implicit) conjecture that for prime p ≡ 1(4), where Bn is the n-th Bernoulli number. This is of interest, if true, since we would have that the middle “B” in the Kummer sequence of B’s is not divisible by p. Confirmed by S. Wagstaff for p 125000, in Math, of Computation 33, (1979).
B.J. Birch and H.P.F. Swinnerton-Dyer. “Notes on Elliptic Curves, II,” J. Reine Angew. Math. 218 (1965), 79–103.
A. Bremner. “A Geometric Approach to Equal Sums of Fifth Powers,” J. Number Theory 13 (1981), 337–354.
S. Chowla. “Contributions to the Analytic Theory of Numbers,” J. Ind. Math. Soo. 25 (1934), 121–126.
S. Chowla. “0n the k-Analogue of a Result in the Theory of the Riemann Zeta-Function,” Math. Zeitsch. 38 (1934), 483–487. Here i is proved that r2 3(n), the_number of representations of n as a sum of three squares is ß(/n log log n) for by means of Square free values proved by Littlewood under the assumption of the “extended” Riemann hypothesis.
S. Chowla and M. Cowles. “The Diophantine Equation 27y2 + 4x3 = M, J. Reine Angew. Math. 291 (1977), 220.
S. Chowla and M. Cowles. “On the Coefficients in the Expansion x (l-xn)2(l-x11n)2 = I? c xn,” J. Reine Angew. Math. 292 (1977), 115–116.
S. Chowla and G. Shimura. “On the Representation of Zero as a Linear Combination of k-th Powers,” Norske Vid. Selsk. Forh. 36 (1963), 169–176.
L.E. Dickson. “History of the Theory of Numbers,” Vol. 2, Diophantine Analysis, Carnegie Institution of Washington (1920). Reprint Chelsea.
P. Erdös. “On the Representation of a Number as a sum of k k-th Powers,” J. Lond. Math. Soo. 9 (1934), 132–136.
E. Landau. Vorlesungen über Zahlentheorie (III), S. Hirzel, Leipzig (1927).
K. Mahler. “On Hypothesis K of Hardy and Littlewood,” J. Lond. Math. Soo. 9 (1934), 136–141.
A. Moessner. “Curious Identit ies,” Scripta Math. 6 (1939), 180.
S. Sastry. “On Sums of Powers,” J. Lond. Math. Soc. 9 (1934), 242–246.
E.S. Selmer. “The Diphantine Equation ax + by J + cz J = 0,” Acta Math. 85 (1951), 203–362.
N.M. Stephens. “The Diophantine Equation x-5 + y = Dz and the Conjectures of Birch-Swinnerton-Dyer,” J. Reine Angew. Math. 231 (1968), 121–162.
A. Tietäväinen. “On a Problem of Chowla and Shimura,” J. Number Theory 3 (1971), 247–252.
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Chowla, S., Cowles, M. (1982). Remarks on Equations Related to Fermat’s Last Theorem. In: Koblitz, N. (eds) Number Theory Related to Fermat’s Last Theorem. Progress in Mathematics, vol 26. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-6699-5_16
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DOI: https://doi.org/10.1007/978-1-4899-6699-5_16
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