Math 350 - Fall 2000 - Number Theory

Sunflowers and the Golden Ratio

1. J. N. Ridley, Packing efficiency in sunflower heads, Math. Biosci. 58 (1982), 129-139.
2. I. Stewart, Life's Other Secret: The New Mathematics of the Living World, J. Wiley, New York, 1998.
3. Websites: http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html, http://www.math.smith.edu/~phyllo/.

Gödel's Incompleteness Theorem and Diophantine Equations

1. H.-D. Ebbinghaus et al., Mathematical Logic (2nd edition), Springer-Verlag, New York, 1994.
2. D. Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid (1st edition), Vintage, New York, 1980.
3. H. J. Keisler and J. Robbin, Mathematical Logic and Computability, McGraw-Hill, New York, 1996.
4. Yu. V. Matiyasevich, Hilbert's Tenth Problem, MIT Press, Cambridge, MA, 1993.
5. R. M. Smullyan, Gödel's Incompleteness Theorems, Oxford Univ. Press, New York, 1992.
6. A. Tarski, Introduction to Logic and to the Methodology of Deductive Sciences (2nd edition), Dover, New York, 1995.

Primality Testing

1. L. M. Adleman, C. Pomerance, and R. S. Rumely, On distinguishing prime numbers from composite numbers, Ann. of Math. (2) 117 (1983), 173-206.
2. A. O. L. Atkin and F. Morain, Elliptic curves and primality proving, Math. Comp. 61 (1993), 29-68.
3. P. Emerson, Prime Number Generation and Primality Testing, Undergraduate Thesis, Middlebury College, 1997, online: http://pantheon.yale.edu/~pre6/.
4. Websites: http://www.utm.edu/research/primes/prove/, http://shemp.optics.rochester.edu:8080/users/stroud/talks/muthukrishnan992/.

The Story of a Diophantine Equation: Elkies' Solution of A⁴ + B⁴ + C⁴ = D⁴

1. N. Elkies, On A⁴+B⁴+C⁴=D⁴, Math. Comp. 51 (1988), 825-835.
2. L. J. Mordell, Diophantine Equations, Academic Press, London, 1984.
3. V. V. Prasolov and Y. Solovyev, Elliptic Functions and Elliptic Integrals (trans. D. Leites), Amer. Math. Soc., Providence, RI, 1997.
4. N. P. Smart, The Algorithmic Resolution of Diophantine Equations, Cambridge Univ. Press, Cambridge, 1998.
5. V. G. Sprindzhuk, Classical Diophantine Equations (trans. from 1982 Russian original), Lect. Notes in Math. 1559, Springer-Verlag, New York, 1993.
6. Website: http://euler.free.fr/faq.htm.

The Origin of Number in Ancient Greece

1. T. Heath, A history of Greek Mathematics, Dover, New York, 1981.
2. G. Johnson, The Arithmetical Philosophy of Nicomachus Of Gerasa, New Era Printing Co., Lancaster, PA, 1916.
3. G. Nicomachus, Introduction to Arithmetic (trans. Martin L. D'ooge), Macmillan, New York, 1926.
4. D. O'Meara, Pythagoras Revived, Clarendon Press, Oxford, 1989.
5. T. Taylor, The Theoretic Arithmetic of the Pythagoreans (1st paperback ed. of 1816 original), Samuel Weiser, York Beach, ME, 1983.

To Leap or not to Leap: The Calendar and the Irrational Number 365.242...

1. N. M. Beskin, Fascinating Fractions (trans. V. I. Kisin), Mir, Moscow, 1980.
2. A. Ya. Khinchin, Continued Fractions (trans. Scripta Technica, Inc.), Dover, Mineola, NY, 1997.
3. D. Steel, Marking Time: The Epic Quest to Invent the Perfect Calendar, J. Wiley, New York, 2000.
4. Websites: http://www.friesian.com/calendar.htm, http://charon.nmsu.edu/~lhuber/leaphist.html.

Fermat, before Wiles

1. A. D. Aczel, Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem, Four Walls Eight Windows, New York, 1996.
2. J. P. Buhler et al, Irregular primes and cyclotomic invariants to four million, Math. Comp. 61 (1993), 151-153.
3. H. M. Edwards, Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory, Springer-Verlag, New York, 1977.
4. D. Marcus, Number Fields, Springer-Verlag, New York, 1977.
5. P. Ribenboim, Thirteen Lectures on Fermat's Last Theorem, Springer-Verlag, New York, 1979.
6. Website: http://www.public.iastate.edu/~kchoi/time.htm.