COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

370 Shen Lin
TABLE 5 P,” c sequence of primes squared
n 1 2 3 4 5 10 15 20
4l 4 9 25 49 121 841 2209 5041
ew 17,163 35,355 124,395 149,403 160,155 269,715 405,003 573,715
ecu
a,-1 = - 4-l
509.858 219.038 132.259
TABLE 6 f(x) = (x2+x)/2, sequence of triangular numbers
n 1 2 3 4 5 100 200 300 400 500 600
S” 1 3 6 10 15 5050 20,100 45,150 80,200 125,250 180,300
e(m 33 50 113 118 173 24,018 90,713 196,133 341,273 532,775 753,774
4.852 4.558 4.373 4.276 4.270 4.194
5.008 4.599 4.529 4.433 4.413
TABLE 7 f(x) = x3
1 2 3 4 5 10 20 30
1 8 27 64 125 1000 8000 27,000
12,758 19,309 23,774 26,861 34,843 80,384 261,517 636,134
TABLE 8 f(x) = x3+ 1
110.266 38.127 26.082
1 2 3 4 5 10 20 30
2 9 28 65 126 1001 8001 27,001
8293 10,387 14,125 17,886 22,331 58,332 222,258 554,195
REFERENCES
79.906 32.377 22.722
1. J. L. BROWN: Note on a complete sequence of integers. American Math. Monthly 68
(1961), 557-560.
2. HANS-EGON RICHERT: Uber Zerlegungen in paarweise verschiedene Zahlen. Nor.&.
Mat. Tiddsskr. 31 (1949), 120-122.
3. K. F. ROTH and G. SZEKERES: Some asymptotic formulae in the theory of partitions.
Q. J. Math. (2) 5 (1954), 241-259.
4. R. L. GRAHAM: Complete sequences of polynomial values. Duke Math. J. 31 (1964),
275-286.
Application of computer to algebraic topology
on some bicomplex manifolds’
HARVEY COHN
1. Introductory remarks. The present calculation is part of a series
concerned with representing the (fundamental) domain of definition of
certain algebraic function fields [l], [2] by computerized geometric visualization.
The ultimate goal is to obtain topological information which
perhaps can be of some value in understanding the algebraic function
fields and some of the number theoretic identities involved.
We are dealing with Hilbert modular functions of two complex variables
over certain real quadratic fields. The theory of algebraic functions of two
complex variables is involved here and the suitability of a representation
such as the Riemann surface is highly questionable in general. We restrict
ourselves to a few carefully chosen cases where IS. B. Gundlach has recently
shown [4], [5] the domain of definition to be representable as a compact
manifold.
In order to visualize a bicomplex space, we must treat four (real) dimensions
to within the limits of three-dimensional intuition. We attempt as
an analogy the visualization of certain (ordinary) modular functions and
their fundamental domains and we show to what limited extent the analogy
can be pursued.
2. Modular group in one variable. Here we consider the upper half plane U
Imz=-0 (2.1)
subject to identifications under the (Klein) modular group G, namely
zo = S(z) = (azf b)/(cz+d), ad-bc = 1 (2.2)
where a, b, c, d are integers. Alternatively, the transformations are represented
by matrices f S where
s = (2.3)
7 Research supported by the U.S. National Science Foundation Grant G-6423 and
computer support contributed by the Applied Mathematics Division of the Argonne
National Laboratory of the U.S. Atomic Energy Commission.
371