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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 (LegendreGauss
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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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 nonintegrality
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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 Spring '11
 Staff
 Number Theory, Polynomials

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