Recently Hagis and McDaniel have studied the largest prime factor of an odd perfect number. Using their results, we begin the study here of the second largest prime factor. We show it is at least 139. We apply this result to show that any odd perfect number not divisible by eight distinct primes must be divisible by 5 or 7.

If N = qkn2 is an odd perfect number, where q is the Euler prime, then we show that σ(n) ≤ qk is necessary and sufficient for Sorli’s conjecture that k = νq(N) = 1 to hold. It follows that, if k = 1 then the Euler prime q is the largest prime factor of N and that q > 10500. We also prove that qk < 23n 2.

City shrinkage is one of the thorny problems that many European cities have to face with nowadays. It is mainly expressed as the decrease of population in these cities. Eastern Germany is one of the pioneers of European shrinking cities with long shrinking history. The paper selects one representative shrinking city Halle (Saale) in eastern Germany as research objective, collecting and investig...

in this work, by employing the krasnosel'skii fixed point theorem, we study the existence of positive solutions of a three-point boundary value problem for the following fourth-order differential equation begin{eqnarray*} left { begin{array}{ll} u^{(4)}(t) -f(t,u(t),u^{prime prime }(t))=0 hspace{1cm} 0 leq t leq 1, &amp; u(0) = u(1)=0, hspace{1cm} alpha u^{prime prime }(0) - beta u^{prime prime...

This paper is concerned with factor left prime factorization problems for multivariate polynomial matrices without full row rank. We propose a necessary and sufficient condition the existence of factorizations class matrices, then design an algorithm to compute all if they exist. implement on computer algebra system Maple, two examples are given illustrate effectiveness algorithm. The results p...