Almost all even positive integers can be represented by the sum of two regular primes. Also one must find the Goldbach Parttion of each even integer as one may think that an even integer is not the sum of two regular primes.

It would be appreciated if someone could find a even number that isn’t the sum of two regular primes.

http://mathworld.wolfram.com/GoldbachConjecture.html

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

http://mathworld.wolfram.com/GoldbachPartition.html

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

Thank you

Nunghead

What is the lower bound for the number of pairwise coprime cubes that can sum to any positive integer.

As always I welcome the readers to try and solve this problem.

Nunghead

The equation n!!+1=k^2n+1 where n>0 has no integer solutions for all positive integers.

Here the double factorial indicates that the product of all even or odd integers =< n.

For example 8!! is equal to 2*4*6*8.

As always I welcome the readers to try proving or disproving this conjecture.

https://en.m.wikipedia.org/wiki/Brocard%27s_problem

Is there a convex shape in any dimension greater than three that has lower packing density than the hypersphere in hyperbolic space.

This question is motivated by Ulam’s Packing conjecture.

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

Nunghead

For any countable comunitive ring is Hilbert’s Tenth problem undecidable and if undecidable is there a minimum set of variables such that the problem is decidable.

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

https://en.m.wikipedia.org/wiki/Hilbert’s_tenth_problem

Nunghead

The generalized Fermat Catalan Conjecture is the statement that the equation:

Does the equation a1^k1+a2^k2+a3^k3… +a^k =b^c  has finitely many or no solutions where a1, a2, a3, … are all relatively prime and that 1/k1 +1/k2+ 1/k3… < 1.

As always I welcome my readers to prove or disprove this conjecture. I would like some help in finding explicit solutions if they exist because I haven’t found an explicit example yet other than Fermat Catalan Conjecture solutions on Wikipedia. The conjecture may be implied by the truth of the “n conjecture” Much help would be appreciated.

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

https://en.wikipedia.org/wiki/Fermat%e2%80%93Catalan_conjecture

Thank you

Nunghead

The equation P, P+2x=Ap +B where all constants are positive integers and all x values are also positive integers,the GCD of (a,b)=1 , b-a if b>a or a-b if a>b /= 2x so that when a prime is prime for both P, P+2x and where x is an even number or odd there also exists two primes of both P,P+2x for Ap+B ,there is infinitely many such primes. For example, take the equation 2p+1. We will explain below how this works. 2p+1=3,3+2(1) where p= (3,5) so 2(3)+1=7 and 2(5)+1=11 where both ( 7,11) are primes. As always, I welcome my readers to try and prove this fact. I will also link the wikipedia articles necessary to understand my conjecture. https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions https://en.wikipedia.org/wiki/Sophie_Germain_prime https://en.wikipedia.org/wiki/Polignac%27s_conjecture https://en.wikipedia.org/wiki/Cousin_prime Thank you Nunghead

There exists infinitely many primes where x is a prime, and ax^n +c is prime for all positive integers, where also all coefficients are positive integers, whose degrees are also all positive integers, wherence the polynomial must also not be factorable over the positive integers, and the GCD of (a,c) must equal one and whose difference between a and c or c and a if c>a or a>c, then a-c or c-a must equal an odd number. This conjecture further generalizes Dirlchlet’s theorem , the Sophie Germain conjecture,and the Bunyakovsky conjecture. To read up more on these conjectures and theorems I have referenced, I have included links to Wikipedia to help you out. As always I welcome the readers to prove this conjecture of mine.

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

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

https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions

Nunghead

I will start by explaining what a transcendental number is and I will also give a short survey article to explain the history of transcendental numbers.

A transcendental number is a number that is not the solution of a polynomial with rational coefficients as opposed to the numbers that are the solutions or roots of the polynomial with rational coefficients, designated by the term algebraic number. Some famous examples of transcendental numbers are Euler’s constant (e) ~=  2.71828… and Pi ~= 3.14159… , famous for being the ratio of a circle’s circumference to its diameter, expressed as the ratio c/d. Now I will present a survey article written by me which will explain the history of the transcendental numbers.

The first recorded appearance of the transcendental numbers in the mathematical literature originates in a paper by Gottfried Leibniz in 1682 where he proves that the trigonometric function sin(x) is not an algebraic function. However it was Euler who gave the modern definition of the transcendental number and  furthermore conjectured that such numbers exist. Later in 1768, John Lambert conjectured that both e and Pi are transcendental. Euler also proved in 1737  that e is irrational however he only published this result seven years later in 1744. In 1844, Joseph Liouville proved that transcendental numbers do exist but still failed to provide an explicit example. In 1850 however, Liouville did construct an infinite series of examples that were transcendental, the most famous of these is known as the Liouville constant. The Liouville constant is the number: 0.110001… . However these are all artificial transcendental numbers and so the jury was still out on whether there were any natural transcendental numbers. In 1873 Charles Hermite proved that e, Euler’s constant, was transcendental, proving that there do exist natural transcendental numbers that don’t have to be constructed for the purpose. However, Pi was yet to be proven as transcendental. In 1882, Ferdinand von Lindermann proved that Pi was transcendental. We shall end here as the rest of the history is few and far between.

Now, I will present my conjecture on a specific transcendental number.
The conjecture is that the number: (n!(1)^-1 u (n+1)!(2) ^-1 u (n+2)!(3)^-1…), where, n=1, is transcendental. Here “u” indicates concatenation of each term with the next term . (1) indicates the degree of the factorial A degree of a factorial indicates the amount of times a factorial of a positive integer >0 is iterated.
One can easily understand this by looking at the first three terms of this series as an example.

(1!(1)^-1 U (1+1)!(2)^-1 U (1+2)!(3)^-1 U …..)

The first few digits of my number is: 1.22601218943565795100204903227081043611191521875016945785727541837850835631156947382240678577958130457082619920575892247259536641565162052015873791984587740832529105244690388811884123764341191951045505346658616243271940197113909845536727278537099345629855586719369774070003700430783758997420676784016967207846280629229032107161669867260548988445514257193985499448939594496064045132362140265986193073249369770477606067680670176491669403034819961881455625195592566918830825514942947596537274845624628824234526597789737740896466553992435928786212515967483220976029505696699927284670563747137533019248313587076125412683415860129447566011455420749589952563543068288634631084965650682771552996256790845235702552186222358130016700834523443236821935793184701956510729781804354173890560727428048583995919729021726612291298420516067579036232337699453964191475175567557695392233803056825308599977441675784352815913461340394604901269542028838347101363733824484506660093348484440711931292537694657354337375724772230181534032647177531984537341478674327048457983786618703257405938924215709695994630557521063203263493209220738320923356309923267504401701760572026010829288042335606643089888710297380797578013056049576342838683057190662205291174822510536697756603029574043387983471518552602805333866357139101046336419769097397432285994219837046979109956303389604675889865795711176566670039156748153115943980043625399399731203066490601325311304719028898491856203766669164468791125249193754425845895000311561682974304641142538074897281723375955380661719801404677935614793635266265683339509760000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000…

I was only able to compute this using Wolfram Alpha for the first three terms however computing more terms of the infinite sequence is unfeasible as Wolfram starts to clam up.

As always, I welcome my readership to give a proof that this number is transcendental. This number was inspired by the Liouville Constant and the Copland-Erdos Constant.

Thank you for reading this article

Nunghead

There is at least one prime in the interval n^a to (n+1)^b, given that,
i) n, a, b are positive integers greater than 1 and cannot have the same value.
ii) a and b are relatively prime; i.e., GCD of (a,b) =1… at least 1 prime exists in this interval.
However, if a and b are not relatively prime, i.e GCD of (a,b) /= 1 then the GCD of the two powers (a,b) results in there being at least greater than 1 amount of primes in the interval.