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Linear Forms in Logarithms

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Diophantine Analysis

Part of the book series: Trends in Mathematics ((TM))

Abstract

Hilbert’s problems form a list of twenty-three problems in mathematics published by David Hilbert, a German mathematician, in 1900. The problems were all unsolved at the time and several of them were very influential for 20th century mathematics. Hilbert believed it was essential for mathematicians to find new machineries and methods in order to solve the mentioned problems. The seventh problem deals with the transcendence of \(\alpha ^\beta \) for algebraic \(\alpha \ne 0,1\) and irrational algebraic \(\beta \). This problem was solved by Gelfond and (independently) Schneider. In 1935, Gelfond found a lower bound for the absolute value of the linear form

$$\varLambda =\beta _1\log \alpha _1+\beta _2\log \alpha _2\ne 0.$$

He proved that

$$\log |\varLambda |\gg -h(\varLambda )^\kappa ,$$

where \(h(\varLambda )\) is logarithmic height of the linear form \(\varLambda \), \(\kappa >5\) and \(\gg \) denotes inequality that is valid up to an unspecified constant factor. He noticed that generalization of his results could prove a huge amount of unsolved problems in number theory.

In 1966 and 1967, in his papers “Linear forms in logarithms of algebraic numbers I, II, III”, A. Baker gave an effective lower bound on the absolute value of a nonzero linear form in logarithms of algebraic numbers, that is, for a nonzero expression of the form

$$\sum _{i=1}^{n}b_i\log \alpha _i,$$

where \(\alpha _1, \dots , \alpha _n\) are algebraic numbers and \(b_1, \dots , b_n\) are integers.

In these notes, we introduce definitions and theorems that are crucial for understanding and applications of linear forms in logarithms. Some Baker type inequalities that are easy to apply are introduced. In order to illustrate this very important machinery, we present some examples and show, among other things, that the largest Fibonacci number having only one repeated digit in its decimal expression is 55, that \(d=120\) is the only positive integer such that the set \(\{d+1, 3d+1, 8d+1\}\) consists of all perfect squares and that some parametric families of \(D(-1)\)-triples cannot be extended to \(D(-1)\)-quadruples.

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Notes

  1. 1.

    Peter Gustav Lejeune Dirichlet (1805–1859), a German mathematician.

  2. 2.

    Adolf Hurwitz (1859–1919), a German mathematician.

  3. 3.

    Theodor Vahlen (1869–1945), an Austrian mathematician.

  4. 4.

    Émile Borel (1871–1956), a French mathematician.

  5. 5.

    Adrien-Marie Legendre (1752–1833), a French matheamtician.

  6. 6.

    Leonhard Euler (1707–1783), a Swiss mathematician.

  7. 7.

    Joseph-Louis Lagrange (1736–1813), an Italian-French mathematician.

  8. 8.

    Joseph Liouville (1809–1882), a French mathematician.

  9. 9.

    Georg Ferdinand Ludwig Philipp Cantor (1845–1918), a German mathematician.

  10. 10.

    Charles Hermite (1822–1901), a French mathematician.

  11. 11.

    Ferdinand von Lindemann (1852–1939), a German mathematician.

  12. 12.

    Karl Weierstrass (1815–1897), a German mathematician.

  13. 13.

    Axel Thue (1863–1922), a Norwegian mathematician.

  14. 14.

    Carl Ludwig Siegel (1896–1981), a German mathematician.

  15. 15.

    Freeman Dyson (1923), an English-born American mathematician.

  16. 16.

    Klaus Friedrich Roth (1925–2015), a German-born British mathematician.

  17. 17.

    Kurt Mahler (1903–1988), a German/British mathematician.

  18. 18.

    OEIS A033307.

  19. 19.

    David Hilbert (1862–1943), a German mathematician.

  20. 20.

    Alexander Osipovich Gelfond (1906–1968), a Soviet mathematician.

  21. 21.

    Theodor Schneider (1911–1988), a German mathematician.

  22. 22.

    Alan Baker (1939), an English mathematician.

  23. 23.

    Eugene Mikhailovich Mateveev (1955), a Russian mathematician.

  24. 24.

    Gisbert Wüstholz (1948), a German mathematician.

  25. 25.

    Harold Davenport (1907–1969), an English mathematician.

  26. 26.

    Andrej Dujella (1966), a Croatian mathematician.

  27. 27.

    Attila Pethő (1950), a Hungarian mathematician.

  28. 28.

    Pierre de Fermat (1601–1665), a French mathematician.

  29. 29.

    Michel Laurent, a French mathematician.

  30. 30.

    Maurice Mignotte, a French mathematician.

  31. 31.

    Yuri Valentinovich Nesterenko (1946), a Soviet and Russian mathematician.

  32. 32.

    Subbayya Sivasankaranarayana Pillai (1901–1950), an Indian mathematician.

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Acknowledgements

The first author, Sanda Bujačić, would like to express her sincere gratitude to Prof. Jörn Steuding for organizing summer school Diophantine Analysis in Würzburg in 2014 and for inviting her to organize the course Linear forms in logarithms. She thanks him for his patience, kindness and the motivation he provided to bring this notes to publishing.

Besides Prof. Steuding, she would like to thank her PhD supervisor, Prof. Andrej Dujella, for his insightful comments during her PhD study, great advices in literature that was used for creating these lecture notes and his constant encouragement. She would also like to thank her co-author, Prof. Alan Filipin, for his kind assistance, guidance, help and excellent cooperation.

Last but not the least, she would like to thank her family: parents, sister and boyfriend for supporting her throughout writing, teaching and her life in general.

Both authors are supported by Croatian Science Foundation grant number 6422.

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Authors and Affiliations

  1. Department of Mathematics, University of Rijeka, Radmile Matejčić 2, 51 000, Rijeka, Croatia

    Sanda Bujačić

  2. Faculty of Civil Engineering, University of Zagreb, Fra Andrije Kačića-Miošića 26, 10 000, Zagreb, Croatia

    Alan Filipin

Authors
  1. Sanda Bujačić
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  2. Alan Filipin
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Correspondence to Sanda Bujačić .

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Editors and Affiliations

  1. Inst. Mathematik, Universität Würzburg Inst. Mathematik, Würzburg, Bayern, Germany

    Jörn Steuding

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Bujačić, S., Filipin, A. (2016). Linear Forms in Logarithms. In: Steuding, J. (eds) Diophantine Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-48817-2_1

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