Download Elementary number theory

Elementary Number Theory
1
Discovering Patterns and Formulas Using Fibonacci Sequences
1
1
2
1
3
2
4
3
5
5
6
8
7
13
8
21
9
34
10
55
11
?
12
?
13
?
14
?
Questions
a) Find a formula for the sum of the first n Fibonacci numbers. For example,
11  2
11 2  3
11 2  3  7
1  1  2  3  5  12
b) If we multiply a Fibonacci number with its second neighbor what do we get? For
example
1 2  2
1 3  3
2  5  10
3  8  24
5  13  65
c) Find the ratios of consecutive Fibonacci numbers. The first few are
n
1
2
3
4
5
6
7
8
9
f(n+1)/f(n) 1
2
1.5
1.666
1.6
1.625
1.61538 1.61904 1.61764
d) Find the values of the following numbers for n=1,2,3,4,5,…. What are these
numbers?
1  5   1  5 
n
n
2n 5
e) Think of a square as 1 and a domino as 2. In how many ways can we make a tile
of given length?
2.
Linear Equations and Highest Common Factor (or Greatest Common Divisor)
a.
b.
c.
3.
Choose three integers a and b. Among all the numbers ax  by what is the
smallest positive integer?
In how many ways can we find the smallest positive number above? Is there a
formula?
Choose three integers a, b, c. Are there integers x and y such that ax  by  c ?
Congruence
a  b(mod m) means a  b is a multiple of m. For example 9  1(mod 4)
a. If x  2(mod 7) , what is x?
b. If x  2(mod 7) , what is x n (mod 7)?
c. What is 2n (mod 7)?
d. What is 2n (mod 6)?
e. True or False  a  b    a 2  b2  (mod 7).
2
f. True or False  a  b    a 2  b2  (mod 6).
2
4.
Prime Numbers
2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 41, 43, ….. are primes
a. Which ones differ by 2? (3 and 5, 5 and 7, 11 and 13, and so on) Find all less than
200.
b. Which ones are n 2  1 ? For example, 2  12  1; 5  22  1; 17  42  1; 37  62  1.
c. What do the primes 5, 13, 17, 29, 37, 41, … share in common? Find 10 more.
d. What do the primes 3, 7, 11, 19, 19, 23, 31, 43, …. share in common? Find 10 more.
e. Which type are more, those in c or d?
f. Every even number greater than 2 is a sum of two primes. Verify this for even
numbers up to 20.
g. True or False: The formula n 2  n  41 gives prime numbers.
h. True or False: The formula n 2  79n  1601 gives prime numbers.
5.
Pythagorean Triples
Three integers  x, y, z  are called Pythagorean Triples (PT) if x 2  y 2  z 2 . For example
3,4,5 are PT
a. Find PT whose common divisor is 1. Such PT are called Primitive PT or PPT
b. True or false: One of PPT is even and the other two are odd.
c. True or false: One of PPT is a multiple of three.
d. Take any four consecutive Fibonacci numbers and do the following:
1) Let x be the product of the first and fourth.
2) Let y be twice the product of the second and third.
3) Let z be the sum of the squares of the second and the third.
Is it true that  x, y, z  is a PT? Is it PPT?
For example, if we take the first four: 1,1,2,3, we get
x  1 3  3
y  2  1 2  4
z  12  22  5
e. Is there a formula that generates PPT?