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4400projects - SOME IDEAS FOR FINAL PROJECTS PETE L. CLARK...

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SOME IDEAS FOR FINAL PROJECTS PETE L. CLARK 1. Project ideas 1.1. Write a computer program that plays Schuh’s divisor game better than you (or I) do. 1.2. Nonunique factorization in the ring R [cos θ, sin θ ] of real trigonometric polynomials. Reference: H.F. Trotter, An Overlooked Example of Nonunique Factorization , Amer- ican Mathematical Monthly 95 (1988), 339–342. http://www.math.uga.edu/ pete/trotter.pdf 1.3. Mordell’s proof of Holzer’s Theorem on Minimal Solutions to Le- gendre’s Equation. Reference. L.J. Mordell, On the Magnitude of the Integer Solutions of the Equation ax 2 + by 2 + cz 2 = 0, Journal of Number Theory 1 (1969), 1–3. http://www.math.uga.edu/ pete/Mordell1968.pdf 1.4. Find all integers of the form x 2 + Dy 2 for D = ± 2 , 3 . 1.5. Find all integers of the form x 2 + 5 y 2 . Remark: This is significantly harder than the cases we looked at in class. Reference: D.A. Cox, Primes of the form x 2 + ny 2 . 1.6. Discuss the history of quadratic reciprocity. References: D.A. Cox, Primes of the form x 2 + ny 2 . Andr´ e Weil, Number Theory: An Approach Through History From Hammurapi to Legendre . 1.7. Discuss the complexity of the following algorithms: powering algo- rithm, Euclidean algorithm, Jacobi symbol algorithm. Reference: Henri Cohen, A Coure in Computational Algebraic Number Theory . 1.8. Investigate algorithms for expressing an integer as a sum of squares. We have determined exactly which integers are sums of two squares. Later on in the course we will prove which integers are sums of four squares (Lagrange’s theorem) and state without proof which integers are sums of three squares (Legendre-Gauss theorem). But these methods do not give efficient algorithms for actually finding any of these representations. E.g, if p 1 (mod 4) is a large prime number, we know that there exist x, y Z such that p = x 2 + y 2 ? There is an obvious trial and error approach to finding x and y : compute p - 1 2 , p - 2 2 , . . . until we get a perfect 1
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2 PETE L. CLARK square. But this is very slow. Find out how to do better! Reference: Henri Cohen, A Coure in Computational Algebraic Number Theory . 1.9. Non/integrality of partial sums of n =1 a n with a n Q . We proved that for all N 2, N n =1 1 n 6∈ Z . There must be similar non-integrality results for other series with rational coefficients, e.g. for partial sums of the p -series n =1 1 n p for p Z + . What interesting results can you find, either by a literature search (and this is one case where I don’t myself know what the literature has to say about this, so it is interesting to me) or by figuring things out yourself? Feel free to change the statement of the problem a little bit if it leads somewhere: e.g. start the sums somewhere other than at
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4400projects - SOME IDEAS FOR FINAL PROJECTS PETE L. CLARK...

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