Problem 3: a) Use Theorem 1 to detmerine the number of integral solutions of the poly- nomial equation a2b223400 11 b) Find all integral solutions (a, b) E Z2 for each of the following three equa- tions a2b21, a2 b2 2 and a2b2 =9 . c) Let q E Z be prime with q a, b E Z with a2 + b2 a, bE Z with a2b2 = q2 3 mod 4. Argue that there are no integers q. Furthermore, show that there are integers d) Consider the subset of skew-symmetric matricies 8{A=(2) e Z2x2det(A) S a where det: Zx2 > Z is the determinant of the matrix. Show that S is closed under matrix multiplication and that the determinat is a homo- morphism from S to Z\ {0}, i.e., det(A B) det(A) det(B) e) Let pE N be a natural number and T {(a,x, y) E Z3a2 + 4xry p and a, z, y > 0} . Show that the set T is finite and (а + 2у, у, х — у — а) if (2х — а, х, а + у — 2) if (a 2x, a y, x) if f(a, x, y) х — у <а< 2x a > 2x is a well-defined map on T to T. f) Show that f is a involution, i.e, (fo f)(a, x, y) (a, x, y). Morover, show that if p is prime and p = 1 mod 4 that f has exactly one fixed point Conclude that #T is odd and further that the involution g: T ->T with (a, x, y)(a, y, x) has at least one fix point. g) Argue carefully that a natural number n E N can be written as a sum of two squares a +b2 if the exponent of every prime number qj with qj = 3 mod 4 is even.
Problem 3: a) Use Theorem 1 to detmerine the number of integral solutions of the poly- nomial equation a2b223400 11 b) Find all integral solutions (a, b) E Z2 for each of the following three equa- tions a2b21, a2 b2 2 and a2b2 =9 . c) Let q E Z be prime with q a, b E Z with a2 + b2 a, bE Z with a2b2 = q2 3 mod 4. Argue that there are no integers q. Furthermore, show that there are integers d) Consider the subset of skew-symmetric matricies 8{A=(2) e Z2x2det(A) S a where det: Zx2 > Z is the determinant of the matrix. Show that S is closed under matrix multiplication and that the determinat is a homo- morphism from S to Z\ {0}, i.e., det(A B) det(A) det(B) e) Let pE N be a natural number and T {(a,x, y) E Z3a2 + 4xry p and a, z, y > 0} . Show that the set T is finite and (а + 2у, у, х — у — а) if (2х — а, х, а + у — 2) if (a 2x, a y, x) if f(a, x, y) х — у <а< 2x a > 2x is a well-defined map on T to T. f) Show that f is a involution, i.e, (fo f)(a, x, y) (a, x, y). Morover, show that if p is prime and p = 1 mod 4 that f has exactly one fixed point Conclude that #T is odd and further that the involution g: T ->T with (a, x, y)(a, y, x) has at least one fix point. g) Argue carefully that a natural number n E N can be written as a sum of two squares a +b2 if the exponent of every prime number qj with qj = 3 mod 4 is even.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Concept explainers
Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.
Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
Topic Video
Question
Problem 3 part b
Transcribed Image Text:Problem 3:
a) Use Theorem 1 to detmerine the number of integral solutions of the poly-
nomial equation
a2b223400
11
b) Find all integral solutions (a, b) E Z2 for each of the following three equa-
tions
a2b21, a2 b2
2 and a2b2 =9 .
c) Let q E Z be prime with q
a, b E Z with a2 + b2
a, bE Z with a2b2 = q2
3 mod 4. Argue that there are no integers
q. Furthermore, show that there are integers
d) Consider the subset of skew-symmetric matricies
8{A=(2)
e Z2x2det(A)
S
a
where det: Zx2 > Z is the determinant of the matrix. Show that S is
closed under matrix multiplication and that the determinat is a homo-
morphism from S to Z\ {0}, i.e., det(A B) det(A) det(B)
e) Let pE N be a natural number and
T {(a,x, y) E Z3a2 + 4xry p and a, z, y > 0} .
Show that the set T is finite and
(а + 2у, у, х — у — а) if
(2х — а, х, а + у — 2) if
(a 2x, a y, x) if
f(a, x, y)
х — у <а< 2x
a > 2x
is a well-defined map on T to T.
f) Show that f is a involution, i.e, (fo f)(a, x, y) (a, x, y). Morover, show
that if p is prime and p = 1 mod 4 that f has exactly one fixed point
Conclude that #T is odd and further that the involution g: T ->T with
(a, x, y)(a, y, x) has at least one fix point.
g) Argue carefully that a natural number n E N can be written as a sum of
two squares a +b2 if the exponent of every prime number qj with qj = 3
mod 4 is even.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 8 steps with 5 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
