Problem 3:a) Use Theorem 1 to detmerine the number of integral solutions of the poly-nomial equationa2b22340011b) Find all integral solutions (a, b) E Z2 for each of the following three equa-tionsa2b21, a2 b22 and a2b2 =9 .c) Let q E Z be prime with qa, b E Z with a2 + b2a, bE Z with a2b2 = q23 mod 4. Argue that there are no integersq. Furthermore, show that there are integersd) Consider the subset of skew-symmetric matricies8{A=(2)e Z2x2det(A)Sawhere det: Zx2 > Z is the determinant of the matrix. Show that S isclosed 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 andT {(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) iff(a, x, y)х — у 2xis 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, showthat if p is prime and p = 1 mod 4 that f has exactly one fixed pointConclude 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 oftwo squares a +b2 if the exponent of every prime number qj with qj = 3mod 4 is even.

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Asked Sep 2, 2019
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Problem 3 part b

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.
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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.

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Expert Answer

Step 1

(1)

consider the first equation:

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a2b 1

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Step 2

Now, since a2 and b2 are always greater than or equal to zero that is:

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a 0 -1sbs1 b=-1,1 similarly b220 -lsas a =-1,1

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Step 3

So,

Here, the possible integral solutions (a, b) for the first equatio...

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