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For all positive integers $$m > n$$, prove that : $$\operatorname{lcm}(m,n)+\operatorname{lcm}(m+1,n+1)>\frac{2mn}{\sqrt{m-n}}$$

Srinivasa Aiyanger Ramanujan

https://primenumberformula.wordpress.com/

According to the Wikipedia article a composite number n is a strong pseudoprime to at most one quarter of all bases below n . Do Fermat pseudoprimes have some similar property ? Is it known what is the largest number of bases for which composite n , that is not Carmichael number is...

Suppose : n \equiv a \pmod 3 ~\text{and}~ a \neq 0 then : n^2 \equiv a^2 \pmod 3 ~\text{and}~ a^2 \neq 0 hence : 3 \nmid n^2 contradiction . Q.E.D.

I asked this question on one another forum but didn't get any answer . Consider the following array of natural numbers : \begin{array}{ccccccccc} 1 & 2 & 4 & 7 & 11 & 16 & 22 & 29 & \ldots \\ 3 & 5 & 8 & 12 & 17 & 23 & 30 & 38 & \ldots \\ 6 & 9 & 13 & 18 & 24 & 31 & 39 & 48 & \ldots...

Hi , I forgot to point out that p has to be greater than three . I know that first two cases are not mutually exclusive . Both values of S_0 can be used in order to prove primality of corresponding Wagstaff number .

\text{Let} ~ W_p ~ \text{be a Wagstaff number of the form :} W_p = \frac{2^p+1}{3}~, \text{where}~p>3 \text {Let's define }~S_0~ \text{as :} S_0 = \begin{cases} 3/2, & \text{if } p \equiv 1 \pmod 4 \\ 11/2, & \text{if } p \equiv 1 \pmod 6 \\ 27/2, & \text{if} ~p \equiv 11 \pmod {12}...

type \displaystyle in front of \prod$$ pr(X=k)=\mu\frac{\displaystyle\prod^{k-1}_{i=1}\{1-\mu+(i-1)\theta\}}{\displaystyle\prod^{k}_{i=1}\{1+(i-1)\theta\}}, \quad \mbox{for k$\geq$ 1,}$$

Hi . You don't have to calculate value of S_{2^{n+1}-3} to find out whether F_n(132) \mid S_{2^{n+1}-3} See Wikipedia article : Lucas-Lehmer primality test One can formulate similar conjectures for other Generalized Fermat numbers . Primality test based on this conjecture written in...

Primality Criteria for F_n(132) \text{Let's define sequence}~ S_i ~\text{as :} S_i= T_{66}(S_{i-1})=2^{-1}\cdot \left(\left(S_{i-1}+\sqrt{S_{i-1}^2-1}\right)^{66}+\left(S_{i-1}-\sqrt{S_{i-1}^2-1}\right)^{66}\right) , ~\text{with}~ S_0=8 \text{and define} ~F_n(132)=132^{2^n}+1 \text{I...

These polynomials are not cyclotomic polynomials. f_n can be rewritten into form : f_n=\displaystyle \sum_{i=0}^n x^{i}+(a-1)\cdot \displaystyle \sum_{i=0}^k x^{i} ,or f_n=\frac{x^{n+1}+(a-1)x^{k+1}-a}{x-1}

( n \equiv 1 \pmod 2 \land n > 1) \Rightarrow \gcd(k-1,3)=1 n \equiv 0 \pmod 2 \Rightarrow \gcd(k+1,3)=1

Is it true that polynomials of the form : f_n= x^n+x^{n-1}+\cdots+x^{k+1}+ax^k+ax^{k-1}+\cdots+a where \gcd(n+1,k+1)=1 , a\in \mathbb{Z^{+}} , a is odd number , a>1, and a_1\neq 1 are irreducible over the ring of integers \mathbb{Z}...

http://en.wikipedia.org/wiki/Proth%27s_theorem