Abstract
We know Pascal’s triangle and planer graphs. They are mutually connected with each other. For any positive integer n, φ(n) is an even number. But it is not true for all even number, we could find some numbers which would not be the value of any φ(n). The Sum of two odd numbers is one even number. Gold Bach stated “Every even integer greater than 2 can be written as the sum of two primes”. Other than two, all prime numbers are odd numbers. So we can write, every even integer greater than 2 as the sum of two primes. German mathematician Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to the golden ratio. We could find the series which is generated by one and inverse the golden ratio. Also we can note consecutive golden ratio numbers converge to the golden ratio. Lothar Collatz stated integers converge to one. It is also known as 3k + 1 problem. Tao redefined Collatz conjecture as 3k − 1 problem. We could not prove it directly but one parallel proof will prove this conjecture.
Highlights
In the west, Pascal’s triangle appears for the first time in the Arithmetic of Jordanus de Nemore (13th century)
In planer graph, an edge draws between two vertices
We can say, an edge draws between two vertices in planer graphs, so only two is acted as the multiplicative factor of binomial expression
Summary
Pascal’s triangle appears for the first time in the Arithmetic of Jordanus de Nemore (13th century). On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII), in which he proposed the following conjecture: Every integer that can be written as the sum of two primes can be written. Goldbach was following the now-abandoned convention of considering 1 to be a prime number, so that a sum of units would be a sum of primes He proposed a second conjecture in the margin of his letter, which is seen to imply the first: Every integer greater than 2 can be written as the sum of three primes. A modern version of Goldbach’s older conjecture of which Euler reminded him is: Every even integer greater than 2 can be written as the sum of two primes. Let us introduce Matrix square concept show which would be the parallel proof for Collatz conjecture and Collatz-Tao conjecture
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