# Linear Forms in Logarithms

• Chapter
• First Online:

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

• 1091 Accesses

## 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.

This is a preview of subscription content, log in via an institution to check access.

## Subscribe and save

Springer+ Basic
\$34.99 /Month
• Get 10 units per month
• 1 Unit = 1 Article or 1 Chapter
• Cancel anytime

Chapter
USD 29.95
Price excludes VAT (USA)
• Available as PDF
• Own it forever
eBook
USD 59.99
Price excludes VAT (USA)
• Available as EPUB and PDF
• Own it forever
Softcover Book
USD 79.99
Price excludes VAT (USA)
• Compact, lightweight edition
• Dispatched in 3 to 5 business days
• Free shipping worldwide - see info
Hardcover Book
USD 79.99
Price excludes VAT (USA)
• Durable hardcover edition
• Dispatched in 3 to 5 business days
• Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

## 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.

## References

1. A. Baker, Linear forms in the logarithms of algebraic numbers, I. Mathematika J. Pure Appl. Math. 13, 204–216 (1966)

2. A. Baker, Linear forms in the logarithms of algebraic numbers, II. Mathematika.J. Pure Appl. Math. 14, 102–107 (1967)

3. A. Baker, Linear forms in the logarithms of algebraic numbers, III. Mathematika J. Pure Appl. Math. 14, 220–228 (1967)

4. A. Baker, Transcendental Number Theory (Cambridge University Press, Cambridge, 1975)

5. A. Baker, H. Davenport, The equations $$3x^2-2=y^2$$ and $$8x^2-7=z^2$$. Quart. J. Math. Oxford Ser. 20(2), 129–137 (1969)

6. A. Baker, G. Wüstholz, Logarithmic forms and group varieties. J. für die Reine und Angewandte Mathematik 442, 19–62 (1993)

7. M. Bennett, On some exponential Diophantine equations of S. S. Pillai. Canad. J. Math. 53, 897–922 (2001)

8. M. Bennett, Rational approximation to algebraic numbers of small height: the Diophantine equation $$| ax^n-by^n |= 1$$. J. Reine Angew. Math. 535, 1–49 (2001)

9. E. Borel, Contribution a l’analyse arithmétique du continu. J. Math. Pures Appl. 9, 329–375 (1903)

10. Y. Bugeaud, M. Mignotte, S. Siksek, Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers. Ann. Math. 163(3), 969–1018 (2006)

11. E.B. Burger, R. Tubbs, Making Transcendence Transparent: An Intuitive Approach to Classical Transcendental Number Theory (Springer, New York, 2004)

12. J.W.S. Cassels, An Introduction to Diophantine Approximation, Cambridge Tracts in Mathematics and Mathematical Physics, vol. 45 (Cambridge University Press, Cambridge, 1957)

13. H. Cohen, Number Theory, Volume I: Tools and Diophantine Equations (Springer, Berlin, 2007)

14. H. Cohen, Number Theory, Volume II: Analytic And Modern Tools (Springer, Berlin, 2007)

15. A. Dujella, The problem of the extension of a parametric family of Diophantine triples. Publ. Math. Debrecen 51, 311–322 (1997)

16. A. Dujella, On the exceptional set in the problem of Diophantus and Davenport. Appl. Fibonacci Numbers 7, 69–76 (1998)

17. A. Dujella, A proof of the Hoggatt–Bergum conjecture. Proc. Amer. Math. Soc. 127, 1999–2005 (1999)

18. A. Dujella, There are only finitely many Diophantine quintuples. J. Reine Angew. Math. 566, 183–214 (2004)

19. A. Dujella, Diofantske jednadžbe, course notes, Zagreb (2006/2007)

20. A. Dujella, Diofantske aproksimacije i primjene, course notes, Zagreb (2011/2012)

21. A. Dujella, C. Fuchs, Complete solution of a problem of Diophantus and Euler. J. London Math. Soc. 71, 33–52 (2005)

22. A. Dujella, A. Pethő, Generalization of a theorem of Baker and Davenport. Quart. J. Math. Oxford Ser. (2) 49, 291–306 (1998)

23. A. Dujella, A. Filipin, C. Fuchs, Effective solution of the $$D(-1)$$- quadruple conjecture. Acta Arith. 128, 319–338 (2007)

24. C. Elsholtz, A. Filipin, Y. Fujita, On Diophantine quintuples and $$D(-1)$$ -quadruples. Monatsh. Math. 175(2), 227–239 (2014)

25. A. Filipin, Linearne forme u logaritmima i diofantska analiza, course notes, Zagreb (2010)

26. A. Filipin, Y. Fujita, M. Mignotte, The non-extendibility of some parametric families of D(-1)-triples. Q. J. Math. 63(3), 605–621 (2012)

27. Y. Fujita, The extensibility of $$D(-1)$$ -triples $$\{1, b, c\}$$. Publ. Math. Debrecen 70, 103–117 (2007)

28. A.O. Gelfond, Transcendental and Algebraic Numbers, translated by Leo F. (Dover Publications, Boron, 1960)

29. B. He, A. Togbé, On the $$D(-1)$$ -triple $$\{1, k^2+1, k^2+2k+2\}$$ and its unique $$D(1)$$ -extension. J. Number Theory 131, 120–137 (2011)

30. M. Hindry, J.H. Silverman, Diophantine Geometry: An Introduction (Springer, New York, 2000)

31. A. Hurwitz, Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche (On the approximation of irrational numbers by rational numbers). Mathematische Annalen (in German) 39(2), 279–284 (1891)

33. M. Laurent, M. Mignotte, Yu. Nesterenko, Formes linéaires en deux logarithmes et déterminants d’interpolation. J. Number Theory 55, 285–321 (1995)

34. F. Lindemann, Über die Zahl $$\pi$$. Mathematische Annalen 20, 213–225 (1882)

35. V.A. Lebesgue, Sur l’impossbilité en nombres entiers de l’équation $$x^m=y^2+1$$. Nouv. Ann. Math. 9, 178–181 (1850)

36. F. Luca, Diophantine Equations, lecture notes for Winter School on Explicit Methods in Number Theory (Debrecen, Hungary, 2009)

37. K. Mahler, Zur approximation der exponentialfunktion und des logarithmus, I, II. J. reine angew. Math. 166, 118–136, 136–150 (1932)

38. K. Mahler, Arithmetische Eigenschaften einer Klasse von Dezimalbruchen. Proc. Kon. Nederlansche Akad. Wetensch. 40, 421–428 (1937)

39. E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers I, II, Izvestiya: Mathematics, 62(4), 723–772 (1998); 64(6), 125–180 (2000)

40. M. Mignotte, A note on the equation $$ax^n-by^n = c$$. Acta Arith. 75, 287–295 (1996)

41. O. Perron, Die Lehre von den Kettenbrüchen (Chelsea, New York, 1950)

42. S. S. Pillai, On $$a^x-b^y=c$$, J. Indian Math. Soc. (N.S.) (2), 119–122 (1936)

43. K.F. Roth, Rational approximations to algebraic numbers. Mathematika 2, 1–20 (1955)

44. J.D. Sally, P.J. Sally Jr., Roots to Research: A Vertical Development of Mathematical Problems (American Mathematical Society, Providence, 2007)

45. W.M. Schmidt, Diophantine Approximation, vol. 785, Lecture Notes in Mathematics (Springer, Berlin, 1980)

46. J. Steuding, Diophantine Analysis (Discrete Mathematics and Its Applications) (Chapman & Hall/CRC, Taylor & Francis Group, Boca Raton, 2005)

47. A. Thue, Über Annäherungswerte algebraischer Zahlen. J. Reine und Angew. Math. 135, 284–305 (1909)

48. T. Vahlen, Über Näherungswerthe und Kettenbr üche, J. Reine Angew. Math. (Crelle), 115(3),221–233 (1895)

## 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.

## Author information

Authors

### Corresponding author

Correspondence to Sanda Bujačić .

## Rights and permissions

Reprints and permissions

© 2016 Springer International Publishing AG