It is proved here that an odd number of the form paS6, where s is square-free, p is a prime which does not divide s, and p and a are both congruent to 1 modulo 4, cannot be perfect. A positive… Expand

In this note, we introduce the notion of the disc induced by an arithmetic function and apply this notion to the odd perfect number problem. We show that under certain special local condition an odd… Expand

In 1888, James Joseph Sylvester (1814-1897) published a series of papers that he hoped would pave the way for a general proof of the nonexistence of an odd perfect number (OPN). Seemingly unaware… Expand

The latter bound of the statement in the title of this paper is improved, showing that the largest prime divisor of an odd perfect number must exceed 10 6 , and Hagis showed that the second largest must exceeds 10 3 .Expand

The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic.Expand

THE third and concluding volume of Prof. Dickson's great work deals first with the arithmetical. theory of binary quadratic forms. A long chapter on the class-number is contributed by Mr. G. H.… Expand