Mathematics 400c
Homework (due May 5)
A. Hulpke
76) For each of the following pairs of Gaussian integers α and β, find Gaussian integers γ
and δ such that α = γβ + δ and N(δ) < N(β):
a) α = 11 + 17i, β = 5 + 3i
b) α = 12 − 23i, β = 7 − 5i
77) Using the classification of Gaussian primes given in the lecture, factorize 6860 = 22 ·5 ·
73 into Gaussian primes. (You might want to compare with the result of the GAP command:
Factors(GaussianIntegers,6860);
78) Represent the prime p = 100049 as sum of two squares, (using the descent method
from the proof in the lecture).
79∗ ) a) Create a list of the divisors of 2925 and determine the number of ways in which
2925 can be written as a sum of two squares.
b) Make a list of all the ways in which 2925 can be written as a sum of squares, check that
it agrees with your count in a)
√
80∗ ) We consider
√ numbers of the form a + b 2−5. 2
a) For α = a + b −5 define the norm N(α) := a + 5b
√ . Show that N(αβ) = N(α)N(β).
b) Show that ±1 are the only units
of
the
form
a
+
b
−5.
√
√
c) Show for each α ∈ {2, 3, 1 + −5, 1 − −5} that α = β · γ implies that β or γ must be a
unit (and thus the numbers
are √
“prime” according to our definition).
√
d) Show that 6 = (1 + −5)(1 − −5) = √
2 · 3 can be written in two different ways as product
of irreducible numbers of the form a + b −5.
Problems marked with a ∗ are bonus problems for extra credit.