The equation n!!(n-1)!!(n-2)!!(n-k)! is equal to y factorial finitely often.

Some explicit examples are 5!!(4!!)(3!!)(2!!)(1!)= 6! , 3!!(2!!)(1!!)=3! , and 7!(6!!)(5!!)(4!!)(3!!)(2!!)(1!!)=10!

Here the double factorial indicates the product of all odd or even integers less than or equal to that integer.

For example, 5!!= 5*3*1 because 5, 3, and 1 are all the odd integers less than or equal to five.

As always I welcome the readers to prove or disprove this conjecture.

Thank you,

Nunghead

If 2+3+5…+n is equal to a prime , then it is also a prime of the form x^2+1 where n is greater than two finitely often?

Some examples will follow

2+3= 2^2+1

2+3+5+7= 4^2+1

2+3+5+7+… 37 =14^2+1

It would be appreciated if someone could find more examples of this phenomenon.

Regarding the composites of the form 2+3+5+7+…n is there only finitely many which sum to triangular numbers. An example of this is 2+3+5=10 , 10 is triangular.

A triangular number is a figurate number that is analogous to the factorial i.e the factorial operation over the positive integers is defined as the multiplication of consecutive positive integers. Similarly the triangular number operation is defined as the successive adding of consecutive positive integers. One can also define a recursive formula for generating triangular numbers e.g n(n+1)|2.

As always I welcome the readers to prove or disprove this conjecture.

Thank you,

Nunghead

x^y=y^x has two more solutions if extended to the Gaussian Integers.

The two other solution of the equation x^y=y^x other than the trivial solutions (4,2) ,(2,4), (-2,4), (-4,2) is the Gaussian Integer pair (4+2i, 4+ 2i) and the conjugate (4-2i, 4-2i).

It would be interesting to expand this identity over any 2^n Cayley-Dickson algebra with dimension greater than one where countable integer subsets are only allowed.

Thank you,

Nunghead

Is there any other Carmichael Numbers other than 1729 that is expressible as a compositorial  number plus one i.e a compositorial number is the factorial of a number divided by the primoreal of the same number. If there is any other such examples they must by finite in number.

https://oeis.org/A036691

https://en.wikipedia.org/wiki/Carmichael_number

Thank you,

Nunghead 

Conjecture no. 34

There is finitely many highly composite numbers such that if one is added to ,it it is a perfect square.

Conjecture no. 35

Is the square root of the highly composite numbers that satisfy conjecture no. 34 always prime if not is it prime infinitely often.

https://en.m.wikipedia.org/wiki/Highly_composite_number

https://oeis.org/A002182

Thank you,

Nunghead

There always exists at least one prime in the interval, Ax and Bx when the GCD of a,b =1 where x is greater than 2. A and B also must be congruent modulo 2 I.e a-b must equal to an even number greater than zero ,a>b.

https://en.m.wikipedia.org/wiki/Bertrand%27s_postulate

Is there arbitrarily long consecutive progressions of Gaussian Primes or any 2^n Cayley Dickson algebras.

Thank you

Nunghead

Does the sums of the reciprocals of figurate numbers equal to a transcendental number infinitely often.

Also, if a regular polygon is not being able to be constructed by a straight edge and compass then does it mean that the pattern of the reciprocals of the objects forming the non constructable polygon summed is transcendental.

https://en.m.wikipedia.org/wiki/Heptagonal_number

https://en.m.wikipedia.org/wiki/Figurate_number
Thank you,

Nunghead

A famous approximation is 22/7 nearly equal to 3.1428571 Another famous approximation is Plato’s approximation sqrt(2)+sqrt(3) nearly equal to 3.14626. Another approximation is the fourth root of 2143/22 given by Ramanujan. Some other approximations are sqrt(10) given by Aryabhatta and 355/113 given by some chinese mathematician. Some of my approximations are sqrt(9.9) correct to two decimal places which is very close to sqrt(2)+sqrt(3), Plato’s approximation and another is sqrt(9.8691) correct to four decimal places. A famous formulae is the Madhava-Leibniz series for pi although it converges very slowly to pi is located below.

1-1/3+1/5-1/7+1/9…= pi/4

These are my calculations I did.

Scratchpad 3  sqrt(10)           = 3.16227766017 355/113             = 3.14159292035 sqrt(2)+sqrt(3)     = 3.14626436994 sqrt(9.9)           = 3.14642654451  sqrt(9.8691)       = 3.14151237464 pi                  = 3.14159265359                                                                             --- https://instacalc.com/53108

https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80

Thank you for reading this blog post,

Nunghead

Conjecture 27. Does 2+3+5+7+…n = x^2 have solutions finitely often. Here are some distinct solutions to this problem down below. Conjecture 28. Is 2+3+5+7+… n /= x^y when y >2 ,where n is the nth prime number. This has been confirmed for all primes upto 10^5 and upto y=10. Conjecture 29. Is x^y – 2+3+5+7+… n =1 for y>1 where n is the nth prime number other than the case (21^2)-2+3+5+7+… 59=1. In the case of n=2 there is finitely many solutions. So for any higher power greater than two is there no solutions i.e the only consecutive powers are the square powers. This has been confirmed upto 10^5.
For all primes upto 10^5 there is only four such solutions for the case n=2.
https://numeracy.car.blog/wp-content/uploads/2019/09/img-20190930-wa0000-158807794.jpg
Conjecture no.30 Is there infinitely many prime sums that sum to a prime infinitely often i.e that 2+3+5+7+…n= a prime infinitely often. Many of these conjectures were inspired by Fermat’s Last Theorem and Catalan’s Conjecture. Thank you, Nunghead