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Squaring the circle:Transcendence, Logarithmic Forms and Diophantine Analysis*

Published online by Cambridge University Press:  01 August 2016

A. Baker
A. Baker*
Affiliation:
DPMMS, 16 Mill Lane, Cambridge CB2 1SB
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Extract

The evolution of transcendence into a fertile theory with numerous and widespread applications has been an especially exciting development of modern mathematics. The subject was originated by Liouville, Hermite and Lindemann during the last century. Liouville showed that there is a limit to the precision by which an algebraic number, not itself rational, can be approximated by rationals and thereby gave the first examples of transcendental numbers; in fact it suffices to take any non-terminating decimal with sufficiently long blocks of zeros or any continued fraction in which the partial quotients increase sufficiently rapidly.

Type
Twentieth Century Mathematics
Information
The Mathematical Gazette , Volume 80 , Issue 487 , March 1996 , pp. 71 - 77
Copyright
Copyright © The Mathematical Association 1996

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Footnotes

*

The text follows a public lecture given at the university of Hong Kong on 31 March 1995.

References

1. Baker, A., Transcendental number theory, Cambridge (1990).Google Scholar
2. Laczkovitch, M., Equidecomposability and discrepancy; a solution of Tarski’s circle-squaring problem, J. reine angew. Math. 404 (1990), pp. 77–117.Google Scholar
3. Wüstholz, G., Zum Periodenproblem, Invent. Math. 78 (1984) pp. 381–391.Google Scholar
4. Wüstholz, G., Transzendenzeigenschaften von Perioden elliptisher Integrale, J. reine angew. Math. 354 (1984) pp. 164–174.Google Scholar
5. Arnol’d, V. I., Huygens, and Barrow, , Newton and Hook, Birkhäuser, Basel (1990).Google Scholar
6. Baker, A. and Wüstholz, G., Logarithmic forms and group varieties, J. reine angew. Math. 442 (1993) pp. 19–62.Google Scholar
7. Steiner, R., On Mordell’s equation y2 - k = x3; a problem of Stolarsky, Math. Computation 46 (1986) pp. 703–714.Google Scholar
8. Stroeker, R. J. and Tzanakis, N., Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms, Acta Arith. 67 (1994) pp. 177–196.Google Scholar
9. Gebel, J., Pethö, A. and Zimmer, H. G., Computing integral points on elliptic curves, Acta Arith. 68 (1994) pp. 171–192.Google Scholar
10. Shorey, T. N. and Tijdeman, L. R., Exponential diophantine equations, Cambridge (1986).Google Scholar
11. Kunrui, Yu, Linear forms in p-adic logarithms III, Compositio Math. 91 (1994) pp. 241–276.Google Scholar
12. Kunrui, Yu and Stewart, C. L., On the abc-conjecture, Math. Ann. 291 (1991) pp. 225–230.Google Scholar
13. Wiles, A., Modular elliptic curves and Fermat’s Last Theorem, Annals of Math. 141 (1995) pp. 443–551.Google Scholar