5. Computational Number Theory - traditional number theory
Prime Numbers
Factors
Counting Factors
D- functions

6. Prime Numbers
One of the most important aspect in number theory which comes into analysis.

7. Number systems
Natural
(0), 1, 2, 3, 4, 5, 6, 7, ..., n
Integers
−n, ..., −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ..., n
Positive integers
1, 2, 3, 4, 5, ..., n
Rational
a⁄b where a and b are integers and b is not zero
Real
The limit of a convergent sequence of rational numbers
Complex
a + bi where a and b are real numbers and i is the square root of −1

8. ℕ Represents the natural numbers. May or may not include zero.
The mathematical symbol for the set of all natural numbers is N, also written ℕ
ℕ⊂ℤ⊂ℚ⊂ℝ⊂ℂ
ℕ Represents the natural numbers. May or may not include zero.
ℤ Represents the integers. (The Z is for Zahlen, which is German for "numbers".)
ℚ Represents the rational numbers. (The Q stands for quotient.)
ℝ Represents the real numbers.
ℂ Represents the complex numbers.

9. The set of integers forms a ring that is denoted ℤ.
One of the numbers ...,-2 ,-1 , 0, 1, 2, ....
The set of integers forms a ring that is denoted ℤ.
A given integer n may be
negative (n ϵ ℤ-),
nonnegative (n ϵ ℤ+),
zero (n = 0), or
positive (n ϵ ℤ+ = ℕ).
A ring in the mathematical sense is a set

10. Prime Numbers
An integer greater than one is called a prime number if its only positive divisors (factors) are one and itself.  
For example, the prime divisors of 10 are 2 and 5; and the first six primes are 2, 3, 5, 7, 11 and 13. 
Positive integers greater than 1 that aren't prime are called composite integers.

11. The First 10,000 Primes (the 10,000th is 104,729)
/ end.

12. The Fundamental Theorem of Arithmetic shows that the primes are the building blocks of the positive integers: every positive integer is a product of prime numbers in one and only one way, except for the order of the factors. (This is the key to their importance: the prime factors of an integer determines its properties.)

13. The ancient Greeks proved (ca 300 BC) that there were infinitely many primes and that they were irregularly spaced (there can be arbitrarily large gaps between successive primes).  
On the other hand, in the nineteenth century it was shown that the number of primes less than or equal to n approaches n/(log n) (as n gets very large); so a rough estimate for the nth prime is n log n
(see the document "How many primes are there?")

14. 1 10 4 2
100 25 3
1,000
168
10,000
1,229 5
100,000
9,592 6
1,000,000
78,498 7
10,000,000
664,579 8
100,000,000
5,761,455 9
1,000,000,000
50,847,534
10,000,000,000
455,052,511 11
100,000,000,000
4,118,054,813 12
1,000,000,000,000
37,607,912,018 13
10,000,000,000,000
346,065,536,839 14
100,000,000,000,000
3,204,941,750,802 15
1,000,000,000,000,000
29,844,570,422,669 16
10,000,000,000,000,000
279,238,341,033,925 17
100,000,000,000,000,000
2,623,557,157,654,233 18
1,000,000,000,000,000,000
24,739,954,287,740,860 19
10,000,000,000,000,000,000
234,057,667,276,344,607 20
100,000,000,000,000,000,000
2,220,819,602,560,918,840 21
1,000,000,000,000,000,000,000
21,127,269,486,018,731,928 22
10,000,000,000,000,000,000,000
201,467,286,689,315,906,290 23
100,000,000,000,000,000,000,000
1,925,320,391,606,803,968,923 24
1,000,000,000,000,000,000,000,000
18,435,599,767,349,200,867,866

17. In 1984 Samuel Yates defined a titanic prime to be any prime with at least 1,000 digits [Yates84, Yates85].  
When he introduced this term there were only 110 such primes known; now there are over 1000 times that many!  And as computers and cryptology continually give new emphasis to search for ever larger primes, this number will continue to grow.  
Before long we expect to see the first fifteen milliondigit prime.

18. Top 10
rank
prime
digits
when 1
2008 2
2009 3 4
2006 5
2005 6 7
2004 8
2003 9
2001 10
19249·
2007
13million digit
$100,000 award

19. Theorem: There are infinitely many prime numbers.
Proof: Suppose the opposite, that is, that there are a finite number of prime numbers. Call them p1, p2, p3, p4,....,pn. Now consider the number
(p1*p2*p3*...*pn)+1
Every prime number, when divided into this number, leaves a remainder of one. So this number has no prime factors (remember, by assumption, it's not prime itself). This is a contradiction. Thus there must, in fact, be infinitely many primes.
So, that proves that we'll never find all of the prime numbers because there's an infinite number of them. But that hasn't stopped mathematicians from looking for them, and for asking all kinds of neat questions about prime numbers.

20. Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers

21. Prime number Set of factors for 5 1 is not a prime number 1 is a unit
For a given instance, say 5, how many divisors are there of 5?
.. How many factors are there?
In the example up .. We illustrate a process of dividing up to 5. dividing by 0 is not allowed (we can approach 0 but not divide by 0)

22. Prime definition
Counting factors are very important for encryption, applying it genetic algorithms
Remember prime numbers always have two factors {1,element}
So for a prime number P
Set of factors for a prime number Dp= {1, P}
|Dp | = 2
The cardanal of the set = number of elements in the set ==2 --- for primary number

23. Factors of a Number Dn ={k1,k2,k3,…..,km} 8 = 23 3×8=23×3 24 =
If we know the K1,k2,..km – we can find the primary power of a decomposition of an integer.
You can write a unique sequence of the products. So that when you multiply them together you get back to the original integer.
WE USE PRIME NUMBERS TO BUILD LARGE NUMBERS

24. 4 = 22 D4={1,2,4} | D4|= 3 Ps = P×𝑃×𝑃…𝑃 S
HOW MANY FACTORS ARE THERE OF AN INTEGER?
The cardinality – the number of elements in a set.
.. Can we discover a pattern to this?
… so let us discuss: is their a pattern in the cardinality of a number.
If you notice the pattern with the prime numbers- these were unique ({1, PrimeNumber})
AND if multiply a sequence of prime numbers we get a composite integer – which in its self will be unique!!
Pi == multiplication as sigma is to summation S

25. relationships

26. Multiplicity function

27. D- function
Tower event
Number of dividers that divide n

28. d(n) – d-function is multiplicatic
Some books have tower symbol

29. 18 = 2×32
D(18)=d(2) ×d(32)
D(2) = 2
d(ps)=s+ 1
d(32)=(2+1)
D(18)=d(2) ×d(32) = 2×3=6

30. 73×5×18 = 2×32×5×73
D(18×5×72)= d(2×32) ×d(5×73)
=d(2) ×d(32) ×d(5) ×d(73)
2×(2+1)×2×(3+1)
=48

31. How many factors does 124 have? 124= (22× 3) 4= 28×3.
It has (8+1)(4+1) = 45 factors
Factor 120 into primes
120 = 60 × 2 = 30× 22= 10 ×3× 22 = 5× 3× 23
Counting factors is very important

32. Greatest Common Factor
Find the GCF(42, 385)
Factors of 42: = 2× 21 =2× 3× 7 (there are = 8 factors)
{1, 2, 3, 6, 7, 14, 21, 42}
Factors of 385: 385=5×77 = 5×7×11
{1, 5, 7, 11, 35, 55, 77, 385}
= 1 =2 =3 =6
= 7
=14
=21
=42
= 1 =5 =7
=35
= 11
=55
=77
=385

33. {1, 2, 3, 6, 7, 14, 21, 42} Ç {1, 5, 7, 11, 35, 55, 77, 385}
= {1, 7}
GCF(42, 385) = 7

34. You can also find the GCF by the prime factorization method
– Find the prime factorization of each number
– Take whatever they have in common (to the highest power possible)

35. Find the GCF(42, 385)
Factorization 42 = 2×21= 2× 3× 7
Factorization 385 = 5×77 = 5×7 ×11
GCF(42, 385) = 71 = 7

36. Find the GCF(338, 507)
Factorization 338 = 2×169 = 2 ×132
Factorization 507 = 3×169 = 3×132
GCF(338, 507) = 132 = 169

46. Prime numbers only

47. Exponents N2 = N × N N3 = N × N × N N4 = N × N × N × N
27 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128
NE × N = N(E + 1)
27 × 2 = 128 × 2 = 256 = 28
NE × N × N = N(E + 2)
NA × NB = N(A + B)
(NA)B = N(A * B)

48. Modulus Arithmetic
...gives the remainder of an integer division instead of the quotient.
For example:
27 % 12 = 3
27 = 3 (mod 12),
or, in words:
  27 is congruent to 3, modulo 12.
In essence, modulus arithmetic consists of taking the infinitely long number-line and coiling it around a finite circle. All the numbers that land on the same point along the circle's edge are considered interchangeable, or congruent.

49. Modulus arithmetic is sometimes called clockface arithmetic -- if it's currently 11 o'clock, then 16 hours later it will be 3 o'clock. (Of course, the analogy is less perfect when the modulus is something other than 12.)
An important feature of modulus arithmetic is that you can replace the terms of an addition operation with congruent values, and still get the right answer:
  16 = 4 (mod 12), therefore   = = 3 (mod 12).

50. Even better, this trick also works with multiplication:
Another example:
9835 = 7 (mod 12), and  
1176 = 0 (mod 12), therefore  
= = 7 (mod 12).
Even better, this trick also works with multiplication:
  9835 × 1176 = 7 × 0 = 0 (mod 12)
(and, if we check, we will see that, yes, 9835  ×  1176 is , and  = 0 (mod 12)).

51. 37 = 7 (mod 10), 287 + 482 = 9 (mod 10), and 895 × 9836 = 0 (mod 10).
If our modulus was 10, then modulus arithmetic would be equivalent to ignoring all but the last digit in our numbers:
  37 = 7 (mod 10),   = 9 (mod 10), and   895 × 9836 = 0 (mod 10).
And, in a sense, a C program does all of its calculations in modulus arithmetic. Since integer calculations in C are permitted to overflow, the high bits silently falling off into the bit bucket, a C program using 32-bit integers is really doing all of its arithmetic modulo 2^32.
As you might imagine, some calculations that are time-consuming and produce huge numbers become trivial in modulus arithmetic. The ability to reduce values to their remainders before doing the actual calculation keeps the calculations from getting out of hand.