Algebra Answers

Problem A.2

Let f(x), g(x) and h(x) be the functions from Problem A.1. Find the derivative λ′(x) of the following function with respect to x:

λ(x) = f(x) · g(x) + f(x) · h(x) − g(x) · h(x)

Problem A.3

The formula for calculating the sum of all natural integers from 1 to n is well-known: Sn =1+2+3+...+n= n2 +n

 2

Similary, we know about the formula for calculating the sum of the first n squares:

n3 n2 n Qn =1·1+2·2+3·3+...+n·n= 3 + 2 + 6

Now, we reduce one of the two multipliers of each product by one to get the following sum: Mn =0·1+1·2+2·3+3·4+...+(n−1)·n

Find an explicit formula for calculating the sum Mn.

2. The functionsf and g are defined as below. f(x) = 3x+2: XER g(x)= 6 2x +3 Find the value of x for which f(g(x)) = 3 Sketch in a single diagram, the graphs of f(x) and f(x). Express each of f(x) and g(x), and solve the equation f¹(x) = g(x)

(a) What are quadratic residues and nonresidues?

(b) Write down the denition of elite primes in mathematical terms.

(c) Prove that

Q2t

i=0

Fi = 222t+1 􀀀 1.

(d) What does the  in the Brun-Titchmarsh inquality represent?

(e) Explicitly show that (x; 2t; 1)  x

2t .

(f) What can you say about the upper bound of E(x) for numbers of the form 32n + 1?

Find the smallest positive integer N that satisfies all of the following conditions:

 N is a square.

 N is a cube.

 N is an odd number.

 N is divisible by twelve different prime numbers.

How many digits does this number N have?

The formula for calculating the sum of all natural integers from 1 to n is well-known:

Sn = 1 + 2 + 3 + .... + n = (n2 + n)/2

Similary, we know about the formula for calculating the sum of the rst n squares:

Qn = 1 .1 + 2 . 2 + 3 .3 + ::: + n .n = n3/3 + n2/2 + n/6

Now, we reduce one of the two multipliers of each product by one to get the following sum:

Mn = 0 .1 + 1 . 2 + 2 . 3 + 3 . 4 + .....+ (n . 1) . n

Find an explicit formula for calculating the sum Mn.

The function is defined by f(x) = ax + b for x ER, where a and b are constants. It is given that f(2)= 1 and f(5) = 7. ii) Solve them and find the values of a and b. 1) Solve the equation f(f(x)) = 0. 1) Set up a pair of simultaneous equations using the information given.

The function is defined by ( ) for , where and are constants. It 

is given that ( ) and ( ) 

i) Set up a pair of simultaneous equations using the information given.

ii) Solve them and find the values of and .

iii) Solve the equation ( ( )) .