By the mean value theorem we have $$\frac{{}^\infty a-{}^{n+1}a}{{}^\infty a-{}^na}=\frac{a^{({}^\infty a)}-a^{({}^na)}}{{}^\infty a-{}^na}=\ln(a^{a^t})\le\ln({}^\infty a)$$ for some $t\in({}^na,{}^\infty a)$. We then have $${}^\infty a-({}^\infty a-{}^{n+1}a)[\ln({}^\infty a)]^x\ge{}^\infty a...

I am new to algebra and help would be more than welcome to tell me if the process I have built is OK, and if my attempt to apply formula on two sets to create a new one is also OK. Context I have a database containing species records (e.g. 10 different species, with 100 rows by species ; in col...

Consider a permutation on an ordered set, $f:\Bbb N_n\to \Bbb N_n,$ s.t. the prime elements assume equal spacing and order after the transformation, and also the composite elements are still ordered from least to greatest after the transformation. Another condition is that prime elements must occ...

For a fixed $n$ and $M$, I am interested in the number of unordered non-negative integer solutions to $$\sum_{i = 1}^n a_i = M$$ Or, in other words, I am interested in the number of solutions with distinct numbers. For $n = 2$ and $M = 5$, I would consider solutions $(1,4)$ and $(4,1)$ equivalen...

How many set of positive integers $\{a,b,c,d\}$ are there such that $ a\leq b\leq c\leq d\leq 50$ and $a+b+c+d=100$? I was thinking about using stars and bars, and it seems to work if there were only three variables: If $a\leq b\leq c\leq 50$ and $a+b+c=100$, then we can define $x=50-a,y=50-b,z...

Seen on http://legauss.blogspot.com.es/2012/05/para-rir-ou-para-chorar-parte-13.html Theorem: $5!/2$ is even. Proof: $5!/2$ is the order of the group $A_5$. It is known that $A_5$ is a non-abelian simple group. Therefore $A_5$ is not solvable. But the Feit-Thompson Theorem asserts that every fi...

In my previous question I asked about the numerical instability and convergence of my tetration. It would seem to be the case that it converges, but suffers from catastrophic cancellation. The definition of my tetration is provided as: $${}^xa=\lim_{n\to\infty}\log_a^{\circ n}({}^\infty a-({}^\i...

As I recently showed in another answer, we have the wonderful pattern: $$\sum_{k=1}^n1=n\implies\int_{-1}^0x~\mathrm dx=\zeta(0)\\\sum_{k=1}^nk=\frac{n(n+1)}2\implies\int_{-1}^0\frac{x(x+1)}2~\mathrm dx=\zeta(-1)\\\sum_{k=1}^nk^2=\frac{n(n+1)(2n+1)}6\implies\int_{-1}^0\frac{x(x+1)(2x+1)}6~\mathr...

A much faster algorithm would be to use bisection. Example code. An even faster algorithm would be to use Newton's method. Example code. An alternative approach would be to use exponentiation and logarithms, rewriting it as $\sqrt[12]2=\exp(\ln(2)/12)$, and then computing those instead. Methods...