Let a and b be positive integers and let d=gcd(a,b) and m=lcm(a,b). Show that if a and b are positive integers, then ab=lcm(a,b)*gcd(a,b). We will denote this integer by lcm(a,b).įor n=8,12,20 and 25, find all positive integers less than n and relatively prime to n.įind integers s and t so that 1=7s+11t. Least Common Multiple: The least common multiple of two nonzero integers a and b is the smallest positive integer that is a multiple of both a and b. Then b=abs+ptb, and since p divides the right side of this equation, p also divides b. Since p does not divide a (and p is prime), a and p are relatively prime so gcd(a,p)=1 and by the previous statement there exist integers s and t such that 1=as+pt. Proof: Suppose p is a prime that divides ab but does not divide a. Thus among all common divisors of a and b, d is the greatest.Įuclid's Lemma:If p is a prime that divides ab, then p divides a or p divides b. Then d=as+bt=(d'h)s+(d'k)t=d'(hs+kt) so that d' is a divisor of d. Now suppose d' is another common divisor of a and b and write a=d'h and b=d'k. This proves that d is a common divisor of a and b.
Analogously (or better yet, by symmetry), d divides b as well. To verify this claim, use the division algorithm to write a=dq+r, where 0<=r 0, then r=a-dq=a-(as+bt)q=a-asq-btq=a(1-sq)+b(-tq) which is in S, contradicting the fact that d is the smallest member of S. Since S is obviously nonempty (if some choice of m and n makes am+bn<0, then replace m and n by -m and -n), the Well Ordering Principle asserts that S has a smallest member, say, d=as+bt. Moreover, gcd(a,b) is the smallest positive integer of the form as+bt. GCD Is a Linear Combination:For any nonzero integers a and b, there exist integers s and t such that gcd(a,b)=as+bt. Since this property cannot be proved from the usual properties of arithmetic, we will take it as an axiom. Legendre and Gauss conjectured, independently, that the number of primes <=n (denoted Π(n))is approximatelyĪn important property of the integers, which we will find useful is the Well Ordering Principle, which states that every set of positive integers contains a smallest member. Thus far the only know Fermat primes are the five that Fermat himself originally found. Euler later discovered that 641 divides F 5, hence it is
However he stopped there because F 5=4,294,967,297. This happens to be true for n=0,1,2.38,39 but fails for n=40 and obviously fails for n=41 (there is a factor of 41 in eachįermat (1601-1665) conjectured that the numbers F n=2 2 n+1 are primes for all n greater that or equal to 0. How can we find prime numbers? Centuries ago, it was thought that if n is an integer then n 2+n+41 is always prime. To begin you will need to acquaint yourself with Cryptography Lesson 2 which includes the concepts of: prime numbers, greatest common divisors, modular arithmetic, etc. Number theory and Cryptography are inextricably linked, as we shall see in the following lessons. See Carl Friedrich Gauss to read more about Gauss.Ĭryptography is the process of transferring information securely, in a way that no unwanted third party will be able to understand the message. Has made it the favorite science of the greatest mathematicians, not to mention its inexhaustible wealth, wherein it so greatly surpasses other parts of mathematics." In number theory prompted Gauss, the "prince of mathematics", to remark that "it is just this which gives the higher arithmetic that magical charm which The great difficulty required to prove relatively simple results
Primes and Prime Factorization are especially important in number theory, as are a number of functions including the Totien function. Number Theory is a vast and fascinating field of mathematics, sometimes called "higher arithmetic," consisting of the study of the properties of whole numbers.
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