Essay On Mathematics


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Introduction

Mathematical inquiry can frequently lead to a jungle of exclusive questions and problems. The field of number theory has a wide assortment of some of these mathematical creatures. These problems may seem to be a bit easy, but they still remain to be dormant for many years with small signs of progress. One such problem is the conjecture of odd perfect numbers for centuries with escaped proof. Perfect numbers are integers that are positive, which denotes their proper divisors sum.  For example, 6 is said to be a perfect number, as its proper divisors sum is 1, 2, and 3. If $ 2^p- 1$ is prime, given that $p$ is prime, then we can state that $2^ {p-1} (2^p-1) $ is a perfect number (Fine, & Rosenberger, 2007).

Are there any odd perfect numbers?
There have been claims that there may never exist an odd perfect number. This is a claim that has never been supported although there are those who think they exist.  Computationally, the estimation has been tested for odd numbers amounting to $10^ {300} $ producing futile results. Many times, mathematicians have given several remarkable results. Eugene Catalan in the year 1888 proved and stated that if there exists an odd perfect number and it is not divisible with 3, 5, or 7, then it is associated with about 26 prime factors.

In his book IX of The Elements, Euclid provided a method for construction of perfect numbers, despite the fact that this method is applicable solitary to even perfect numbers. In a letter dating back to 1638 written to Mersenne, Descartes projected that any perfect number relates to Euclid’s form, and said that he identified no reason why there could never exist an odd perfect number. Descartes is, however, one of the first people to consider odd perfect number existence. Up to this date, it is not clearly known whether there exists any odd perfect number, although numbers that range from 10˄300 have been tested without any success. Euclid knew that if 2p-1(2p-1) is a perfect amount, then 2p-1 is a prime. This implies that if 2p-1 is prime, then p is also said to be prime, but not the contrary (Fleming, 2003).
If odd perfect number exists, then N must fulfill the following circumstances:

N > 101500.
N cannot be divided by 105.
N is of the procedure N ≡ 1 (mod 12), N ≡ 117 (mod 468), or N ≡ 81 (mod 324).
 
Where:
q, p1, ..., pk are different primes as stated by Euler.
q ≡ α ≡ 1 (mod 4) as stated by Euler.
The least prime factor of N is less than (2k + 8) / 3.
Either qα > 1062 or p j2ej > 1062 for some j.

The major prime factor of N is bigger than 108.

The second chief prime factor is more than 104, and the third main prime factor is larger than 100.

N has about 101 prime factors and about 9 different prime factors. If 3 are not among the factors of N, then N has about 12 diverse prime factors.

How many even perfect numbers are there? Finitely many or infinitely many? If finite, how many?

The arithmetic fundamental theorem depicts that each positive integer can be factored into prime products in a very unique way.  Furthermore, each finite prime product number is said to be equals to some positive integer. This states that there is a one-to-one correspondence between finite prime products and positive integers. This leads us to understanding positive integers by disintegrating them to prime factors and reviewing them just like the way we try to understand molecules by reviewing the atoms they are made of. The more one tries to find for integers, which are prime, then we go ahead to understand that there are smaller proportions of integers, which are positive. Euclid stated that there may be many infinitely many primes. Suppose we state that there are maybe finitely many primes, like k of which we can denote by 2 =p1 Are there infinitely many twin primes?

A twin prime is said to be a prime number that varies by two from another prime number. Excluding the pair (2, 3), this is said to be the smallest likely variance among the two primes. Various examples of twin primes include (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), and (821, 823) among others.The question depicting the existence of noticeably many twin primes is one open question associated with the number theory.The twin prime conjecture content states that there are noticeably many prime p to an extent that p+2 is said to be prime. De Polignac in 1849 completed the broader conjecture that for any natural number k, there are noticeably a lot of prime pairs like p and p′ depicting that p′ − p = 2k. The situation k = 1 is depicted as a twin prime conjecture (Fine, & Rosenberger, 2007).

All positive, even integer’s ≥4 can be expressed as the sum of 2 primes.

 
The Gold Bach original conjecture, which is also known as the ternary was a letter written to Euler in the year 1742 stating that there may be possibilities that every number which is greater than 2 can be the sum of three primes. As a matter of fact, Gold Bach considered 1 to be a prime, an aspect no longer followed. As explained by Euler, all positive even integers may be expressed as the totality of two primes. A gold Bach number is said to be an integer, which is even and positive, and which is expressed as the totality of two primes. Gold Bach conjecture claims that all integer, which are even and greater than or may equate to 4 are depicted to be Gold Bach numbers (Crandall, & Pomerance, 2001). Gold Bach partition is the expressions of even numbers, amounting to the totality of two primes. These partitions may include

4 = 2 + 2
6 = 3 + 3
8 = 3 + 5
10 = 3 + 7 = 5 + 5

The Collatz Conjecture.
Named after Lothar Collatz, the Collatz conjecture is an unsolved conjecture proposed in 1937. This conjecture is also called the Hasse’s algorithm, or the Syracuse problem. It states that involved sequence numbers are called the hailstone numbers. Use one natural number n. If say n is even, divide it by 2 and get n/2, if n is odd, then multiply it by 3 then add 1 to get 3n+1. Recap the process indeterminately. The conjecture states that no matter the number you start with, eventually you will reach to 1. This property is also called oneness (Fleming, 2003).
 

References

Crandall, R. E., & Pomerance, C. (2001). Prime numbers: A computational perspective. New York: Springer.
Fine, B., & Rosenberger, G. (2007). Number theory: An introduction via the distribution of primes. Boston: Birkhäuser.
Giordano, P. (2010). The solitude of prime numbers. New York: Pamela Dorman Books/Viking.
Fleming, D. K. (2003). How to prove the Collatz conjecture. S.l: lulu.com.