Number Theory

xviiiTable of Contents11.3.4 χ-Power Sums Involving p-adic Logarithms . . . . . . . . . . 92211.3.5 The Function L p (χ, s) Around s = 1 . . . . . . . . . . . . . . . . 92811.4 Applications of p-adic L-Functions . . . . . . . . . . . . . . . . . . . . . . . . 93111.4.1 Integrality and Parity of L-Function Values . . . . . . . . . . 93111.4.2 Bernoulli Numbers and Regular Primes . . . . . . . . . . . . . 93511.4.3 Strengthening of the Almkvist–Meurman Theorem . . . 93811.5 p-adic Log Gamma Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94011.5.1 Diamond’s p-adic Log Gamma Function . . . . . . . . . . . . . 94111.5.2 Morita’s p-adic Log Gamma Function . . . . . . . . . . . . . . . 94711.5.3 Computation of some p-adic Logarithms. . . . . . . . . . . . . 95711.5.4 Computation of Limits of some Logarithmic Sums . . . . 96711.5.5 Explicit Formulas for ψ p (r/m) and ψ p (χ, r/m) . . . . . . . 97011.5.6 Application to The Value of L p (χ, 1) . . . . . . . . . . . . . . . . 97311.6 Morita’s p-adic Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . 97611.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97611.6.2 Definitions and Basic Results . . . . . . . . . . . . . . . . . . . . . . 97611.6.3 Main Properties of the p-adic Gamma Function . . . . . . 98111.6.4 Mahler–Dwork Expansions Linked to Γ p (x) . . . . . . . . . . 98611.6.5 Power Series Expansions Linked to Γ p (x) . . . . . . . . . . . . 98911.6.6 The Jacobstahl–Kazandzidis Congruence . . . . . . . . . . . . 99211.7 The Gross–Koblitz Formula and Applications . . . . . . . . . . . . . . 99511.7.1 Statement and Proof of the Gross–Koblitz Formula . . . 99511.7.2 Application to L ′ p (χ, 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100011.7.3 Application to the Stickelberger Congruence . . . . . . . . . 100111.7.4 Application to the Hasse–Davenport Product Relation 100311.8 Exercises for Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007Part IV. Modern Methods12. Applications of Linear Forms in Logarithms . . . . . . . . . . . . . .102112.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102112.1.1 Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102112.1.2 Applications to Diophantine Equations and Problems . 102312.1.3 A List of Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102412.2 A Lower Bound for |2 m − 3 n | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102512.3 Lower Bounds for the Trace of α n . . . . . . . . . . . . . . . . . . . . . . . . 102912.4 Pure Powers in Binary Recursive Sequences . . . . . . . . . . . . . . . . 103012.5 Greatest Prime Factors of Terms of Some Recursive Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103112.6 Greatest Prime Factors of Values of Integer Polynomials . . . . . 103212.7 The Diophantine Equation ax n − by n = c . . . . . . . . . . . . . . . . . . 103312.8 Simultaneous Pell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103412.8.1 General Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103412.8.2 An Example in Detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035