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.
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
Problem 3 part g
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