The first two series for e are of my own innovation are far faster than Newton’s method below giving 12+ digits after 7 terms compared to 15+ terms for Newton’s series. Nunghead
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A criterion for twin primes
This criterion is based on Wilson’s Theorem. Recall that Wilson’s Theorem states that a number (x) is prime if and only if (x-1)!+1)/x is an integer. The generalization for twin primes goes like this If x is the lesser member of a twin prime pair then (x-1)!+1)/x and (x-1)!+1)/(x+2) are both integers therefore the denominators …
On Polygonal Numbers-A formula
A formula for iterating the nth polygonal number is below. In the formula below a is herewith denoted to be greater than zero. For A=1 one gets the triangular numbers, a=2 the square numbers, and so on. This formula was derived when the author observed that the triangular numbers can be represented by the sum …
A new Primality Proving Algorithm
The algorithm works like this. First, we start with the equation x+y=z (x,y)>0 Next, we pick an input value for z. In this case we will pick z= 5. Then, we proceed to solve the equation in terms of x. Next, we enumerate the solutions to the equation. Namely, 4+1=5, 3+2=5, 2+3=5, and 1+4=5. Then, …
Sin(x) Formula
The following formula of mine is a rediscovery of the standard taylor series for the sin function. I thought to note this because of its impact on my mathematical development as an individual. At first I had only discovered the Taylor Series of sin(1) that being (down below on the right side) where x=1 while …
A proof that the sum of two consecutive triangular numbers is square.
Without further ado we begin. Take a look at the sequence 0, 1, 3, 6, 10, 15 ,21,… Sum every two consecutive terms in the sequence. What do you notice? The sequence is the square numbers I hear you say. And indeed they seem to be the proverbial square numbers i.e 1, 4, 9 ,16, …
Continue reading “A proof that the sum of two consecutive triangular numbers is square.”
Question 7.0 A question on a variant of Carmichael’s Conjecture.
A announcement
It seems that I have rediscovered the centered polygonal numbers. Just a little announcement for all of you. This announcement refers to previous post on series of difference 2ax. The formula for the centered polygonal numbers is A(x(x+1)/2)+1. A represents the constant multiplier between every two consecutive terms. Thank you, Nunghead https://en.m.wikipedia.org/wiki/Centered_polygonal_number#
Conjecture no. 46 On a special pair of twin primes.
Is there a pair of prime numbers such that both n x 2^n +1 and n x 2^n -1 are prime if n is also prime? I conjecture that there is not such a pair. As always I welcome the readers to prove or disprove this conjecture. https://en.wikipedia.org/wiki/Woodall_number https://en.wikipedia.org/wiki/Cullen_number Thank you, Nunghead
An interesting Formula
A(x(x+1)/2))+1 where a is a even number; x is any number from 1 to infinity. This strikingly general, simple, formula subsumes all my previous formulas for finding the rules of series like 1, 5, 13, 25,41,61, and so on… with the exception of series that do not have difference ax where a >1. I will …
My Mathematics work and other extraneous diversions 