Using a new factor chain argument, we show that 5 does not divide an odd perfect number indivisible by a sixth power. Applying sieve techniques, we also find an upper bound on the smallest prime divisor. Putting this together we prove that an odd perfect number must be divisible by the sixth power of a prime or its smallest prime factor lies in the range 108 < p < 101000. These results are gene...

This paper investigates the randomness and cryptographic properties of the Narayana series modulo p, where p is a prime number. It is shown that the period of the Narayana series modulo p is either p2+p+1 (or a divisor) or p2-1 (or a divisor). It is shown that the sequence has very good autocorrelation and crosscorrelation properties which can be used in cryptographic and key generation applica...

In this paper, it is shown that the value semigroup of a prime divisor of the second kind on a 2-dimensional regular local ring is symmetric. Further, a necessary and sufficient condition for two prime divisors of the second kind on a 2-dimensional regular local ring to have the same value semigroup is obtained.

We consider very short sums of the divisor function in arithmetic progressions prime to a fixed modulus and show that “on average” these sums are close to the expected value. We also give applications of our result to sums of the divisor function twisted with characters (both additive and multiplicative) taken on the values of various functions, such as rational and exponential functions; in pa...

This paper discusses the additive prime divisor function A(n) := ∑ pα||n αp which was introduced by Alladi and Erdős in 1977. It is shown that A(n) is uniformly distributed (mod q) for any fixed integer q > 1.

Numbers that are not prime are defined as composite. From this definition, we can see that a prime number can never be even, since it would have the divisor of 2. Of course, the only exception to that rule is the prime number 2, which has no other divisors but itself. Therefore, we can extend the definition to say that every prime number greater than 2 must be odd. From the above definition, we...

For each proper divisor d of (q − √q + 1), q being a square power of a prime, maximal curves Fq-covered by the Hermitian curve of genus 1 2 ( q− √ q+1 d − 1) are constructed.