NUMBER THEORY
ZHIYONG YAN
Abstract. Three proofs of The fundamental theorem of arithmetic theorem;
sum of integers;
Contents
1. The fundamental theorem of arithmetic theorem
2. Sum of integers
3. Prime
4. Equation
5. Number
Appendix A. The endoscopic correspondences
A.1. The endoscopic correspondences
A.2. The endoscopic correspondences
Appendix B. The endoscopic correspondences
References
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There days I am reviewing Elementary Number Theory. Of course the most
excellent book about this subject are [1, Th. 2] and [2].
1. The fundamental theorem of arithmetic theorem
1.1. Theorem (The fundamental theorem of arithmetic theorem). Every positive
integer n greater than 1 can be expressed as a product of primes in one way only,
apart from rearrangement of factors.
The theorem will be used in Subsection. 2.
Proof. Two proofs can be found at http://en.wikipedia.org/wiki/Fundamental_
theorem_of_arithmetic.
Third proof is as follows:
Let n be the least number which can be factorized into primes in more than one
way, then the same prime P cannot appear in two different factorizations of n. So,
we have
n = p1 p2 ... = q1 q2 ...,
where the p and q are primes, no p is a q and no q is a p.
Take p1 to be the smallest p, then p21 ≤ n; Similarily, suppose q1 is the least q,
we have q12 ≤ n. It is important ot show that p1 q1 |n.
2000 Mathematics Subject Classification. Primary 14H60, 20G35; Secondary 14F20, 14K30.
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ZHIYONG YAN
1.2. Remark. The first proof uses The remainder for natural numbers; on the
contrary, it seems that the third proof has nothing to do with the theorem and it
is dependent on the following lemma:
Lemma (Least Number Principle). Any non-empty subset of N has a least number.
And second proof uses both. This is very interesting. 2. Sum of integers
2.1. Theorem. a, b are integers, (a, b)= 1. p is an odd prime divisor of a2 + b2 .
then p ≡ 1(mod4).
3. Prime
The Prime theorem LATEX
4. Equation
use Theorem 1.1.
5. Number
five
Appendix A. The endoscopic correspondences
A.1. The endoscopic correspondences.
A.2. The endoscopic correspondences.
Appendix B. The endoscopic correspondences
References
[1] G.H.Hardy and E.M.Eright, An introduction to the theory of numbers(Sixth edition). Oxford,
2008. available at http://www.whu.edu.cn/.
[2] Hua Luogeng, An introduction to the number theory. Beijing: Science Press, 1957.
Department of Mathematics, Harvard University
E-mail address: newnumbertheory@gmail.com