On the Diophantine Equation Fn = x^a \pm x^b \pm 1 in Mersenne and Fermat Numbers

Mersenne and Fermat Numbers

The first seventeen even perfect numbers are therefore obtained by substituting these values of ra in the expression 2n_1(2n —1). The first twelve of the Mersenne primes have been known since 1914; the twelfth, 2127 —1, was indeed found by Lucas as early as 1876, and for the next seventy-five years was the largest known prime. More details on the history of the Mersenne numbers may be found in ...

Pm Numbers, Ambiguity, and Regularity

We introduce the pseudo-m-ary (Pm) number system in which numbers are represented by sums of the form ∑ i≥0 ai(m i+1 − 1). We characterize the Pm representations that are produced by the greedy algorithm and show that they form a regular set. In addition, we show that the set of Pm representations that are the sole representations for their corresponding numbers is also a regular set.

Two-Generator Numerical Semigroups and Fermat and Mersenne Numbers

Given g ∈ N, what is the number of numerical semigroups S = 〈a, b〉 in N of genus |N \ S| = g? After settling the case g = 2 for all k, we show that attempting to extend the result to g = p for all odd primes p is linked, quite surprisingly, to the factorization of Fermat and Mersenne numbers.

On the equiponderate equation xa+xb+x=xc+xd+1 and a representation of weight quadruplets

On pm$^+$ and finite character bi-amalgamation

‎Let $f:Arightarrow B$ and $g:A rightarrow C$ be two ring homomorphisms and let $J$ and $J^{'}$ be two ideals of $B$ and $C$‎, ‎respectively‎, ‎such that $f^{-1}(J)=g^{-1}(J^{'})$‎. ‎The bi-amalgamation of $A$ with $(B,C)$ along $(J,J^{'})$ with respect of $(f,g)$ is the subring of $Btimes C$ given by‎ ‎$Abowtie^{f,g}(J,J^{'})={(f(a)+j,g(a)+j^{'})‎/ ‎a in A‎, ‎(j,j^{'}) in Jtimes J^{'}}.$‎ ‎In ...