Download Pythagoras Pythagoras A right triangle, such as shown in the figure

Pythagoras
Pythagoras
A right triangle, such as shown in the figure below, has one 90◦
angle. The long side of length c is the hypotenuse. The short
leg (or cathetus) has length a, and the long leg has length b.
The theorem of Pythagoras, which is one of the most important
in mathematics because of its many applications, states that
a2 + b2 = c2 .
There are many proofs of this theorem. An elegant one only
requires that we know that the area of a square of side L is L2
2
a + b2 = c 2
c
b
Equal Areas
a
Area = c
c
c
2
2
Area = a + b
b
c
b
a
b!
2
b
a
a
c
a
b
• Pythagorean triplets
A famous triplet is a = 3, b = 4, c = 5 for which 32 + 42 = 52 .
Another one is a = 5, b = 12, c = 13 for which 52 + 122 = 132 .
Number theory is the branch of mathematics that deals with
whole numbers. Above we saw two examples of whole number
solutions of the equation a2 +b2 = c2 . Are there infinitely many
solutions? The answer is yes. One can construct infinitely
many solutions by letting
a = m2 − n2 ,
and c = m2 + n2 .
b = 2mn,
For example, n = 1, m = 2 give a = 3, b = 4, c = 5,
n = 1, m = 4 give a = 15, b = 8, c = 17,
n = 2, m = 3 give a = 5, b = 12, c = 13,
n = 2, m = 5 give a = 21, b = 20, c = 29, and so on.
Fermat’s last theorem
A famous question in number theory is whether there are any
whole number solutions to the equation a3 + b3 = c3 . The
answer is no. In 1637 Pierre de Fermat stated without proof
that no whole number triplets exist that satisfy ak + bk =
ck where k is any whole number k = 2, 3, 4, ..... The best
mathematicians tried to prove this without success until 1994
when it was finally proved by Andrew Wiles.
Rational and Irrational numbers
The integers, or whole numbers, are ...-2,-1,0,1,2,3... Rational
numbers are those that can be written as the ratio or quotient
of two whole numbers,
such as 2/3, 3/2, 1/2, 5/1, -3/4,... The
√
square root of 2, 2 is a number that when squared gives 2,
2
√
2 = 2.
√
The Pythagoreans were upset to discover that 2 is irrational,
it cannot be written as the ratio of two whole numbers. The
proof is as follows. Let
√
2=
p
.
q
where p and q, are two whole numbers that have no common
factors. If initially they had common factors, cancel them. In
particular, p and q are not both even. I will show this is not
possible.
2
2
√
p
p2
2 =2=
= 2 , so p2 = 2q 2 , therefore p2 is even.
q
q
Since p2 is even, we see that p is even, so we can write p = 2m,
where m is another whole number. We will show that q is also
even. Since
p2 = 2q 2 and p = 2m, we get (2m)2 = 4m2 = 2q 2 , or 2m2 = q 2 .
But this means that q is also even, which is a contradiction.
Prime Numbers and Public-key Encription
A prime number (or a prime) is a natural number (1,2,3,4,5,...)
that has exactly two distinct natural number divisors: 1 and
itself. The smallest twenty-five prime numbers (all the prime
numbers under 100) are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,
41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. Euclid proved
that there is an infinite number of primes. RSA encryption
(which stands for Rivest, Shamir and Adleman, three mathematicians who first publicly described it) involves a public key
and a private key. The public key can be known to everyone
and is used for encrypting messages. Messages encrypted with
the public key can only be decrypted using the private key.
The keys for the RSA algorithm are generated the following
way (Wikipedia RSA Algorithm):
(1) Choose two distinct prime numbers p and q that are kept
secret.
• For security purposes, the integers p and q should be chosen
at random, and should be of similar bit-length. Prime integers
can be efficiently found using a primality test.
(2) Compute n = pq.
• n is used as the modulus for both the public and private keys
(3) Compute φ(n) = (p − 1)(q − 1).
(4) Choose an integer e such that 1 < e < φ(n) and the greatest
common factor of e and φ(n) is 1. For example if e = 1 × 2 × 4
and φ(n) = 1 × 3 × 5, the greatest common factor is 1.
• The public key consists of n and the public exponent e. Sometimes e = 3 is used.
(5) determine d such that d × e = 1modφ(n). This means that
d × e divided by φ(n) has a remainder of 1. For example with
e = 8 and φ(n) = 15, one needs d = 2.
d is kept as the private key exponent. The public key consists
of the modulus n and the public (or encryption) exponent e.
The private key consists of the private p, q, and exponent d
which must be kept secret.
Alice transmits her public key (n,e) to Bob and keeps the private key secret. Bob then wishes to send message M to Alice.
He first turns M into an integer 0 < m < n by using an agreedupon reversible protocol known as a padding scheme. A trivial
one would be to just substitute every character by its ASCII
code. He then computes the ciphertext c corresponding to
c = me (mod n).
Bob then transmits c to Alice.
Alice can recover m from c by using her private key exponent
d via computing m = cd (mod n). Given m, she can recover
the original message M by reversing the padding scheme.
Fermat’s Little Theorem The whole thing depends on Fermat’s Little Theorem, which states that if p is a prime number,
then for any integer a, ap − a is a multiple of p, which in terms
of modular arithmetic says ap = a(mod p). If a is not a multiple
of p, then ap−1 = 1(mod p)