Finally

Finally my statements are as below :-

1.

((n+1)^m/(n+2)^m-(n-1)^m/n^m)/(((n+1)/(n+2)-(n-1)/n)^m)=((n(n+1))^m-((n-1)(n+2))^m)/2^m.

It is always a positive integer for positive integer m and n.

2.

((n+1)^m/(n+2)^m-(n-1)^m/n^m)/(((n+1)/(n+2)-(n-1)/n)^m)=0

and for m equal to 2 gives -1.618033 . . . (the golden ratio)

3.

((n+1)^m/(n+2)^m-(n-1)^m/n^m)/(((n+1)/(n+2)-(n-1)/n)^m)+42 and for m equal to 2 is a fairly good prime number generator.

4.

((n+1)^m/(n+2)^m-(n-1)^m/n^m)/(((n+1)/(n+2)-(n-1)/n)^m) can be approximated to a/((n+1)^2)-b/((n+1)^3)+c/((n+1)^4)-d/((n+1)^5).

Where a,b,c and d are in Arithmetic Progression. First term is 2m. Common ratio is 2m(m-2).

5.

((n+1)^m/(n+2)^m-(n-1)^m/n^m)/(((n+1)/(n+2)-(n-1)/n)^m) traverses across number series, squared series, triangle series in a beautiful way.(Difficult to explain in words but can be demonstrated)

6.

I believe this to be the law of four fingers. Four fingers play a crucial role of approximation in human thought. And also accuracy.

7.

((n)^m/(n+1)^m-(n-1)^m/n^m)/(((n)/(n+1)-(n-1)/n)^m)=((n)^2m-((n-1)(n+2))^m)

It is always a positive integer for positive integer value of m and n.

8.

((n)^m/(n+1)^m-(n-1)^m/n^m)/(((n)/(n+1)-(n-1)/n)^m) traverses across number series, squared number series, triangle series in a beautiful way. ( Difficult to explain in words but can be demonstrated)

9.

I believe statement 7 is law of three fingers. Only accuracy. No approximations (As far as I tried, there were no general statement on approximations (the way there was in four fingers)).

10.

((n+1)^m/(n+2)^m+(n-1)^m/n^m)/(((n+1)/(n+2)-(n-1)/n)^m) also give positive integer and so does ((n)^m/(n+1)^m+(n-1)^m/n^m)/(((n)/(n+1)-(n-1)/n)^m) for positive integer value of m and n. 

Cats and dogs have 1+4 toes in their front feet. And 1+3 in their hind feet.

Monkeys, apes and humans have 1+4 and 1+4.

Birds 1+3.

Yesterday, I saw the video of a simpler solution to quadratic equations by Po-Shen-Loh of Carnegie Mellon University. I liked it so much. It was much easy than the traditional formula. Po-Shen-Loh had submitted this paper in 2019.

My good wishes to all my blog readers !!