Two-Generator Numerical Semigroups and Fermat and Mersenne Numbers
نویسندگان
چکیده
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.
منابع مشابه
Fast algorithms for 2-D circular convolutions and number theoretic transforms based on polynomial transforms over finite rings
In this paper, we develop new fast algorithms for 2-D integer circular convolutions and 2-D number theoretic transforms (NTT). These new algorithms, which offer improved computational complexity, are constructed based on polynomial transforms over 2, ; these transforms are Fourier-like transforms over Z1,[.r], which is the integral domain of polynomial forms over Z,. Having defined such polynom...
متن کاملOn the Norms and Spreads of Fermat, Mersenne and Gaussian Fibonacci RFMLR Circulant Matrices
Abstract: In this paper, we consider norms and spreads of RFMLR circulant matrices involving the Fermat, Mersenne sequences and Gaussian Fibonacci number, respectively. Firstly, we reviewed some properties of the Fermat, Mersenne sequences, Gaussian Fibonacci number and RFMLR circulant matrices. Furthermore, we give lower and upper bounds for the spectral norms and spread of these special matri...
متن کاملOn the Nonexistence of Odd Perfect Numbers
In this article, we show how to prove that an odd perfect number with eight distinct prime factors is divisible by 5. A perfect number N is equal to twice the sum of its divisors: σ(N) = 2N . The theory of perfect numbers when N is even is well known: Euclid proved that if 2 − 1 is prime, then 2p−1(2p − 1) is perfect, and Euler proved that every one is of this type. These numbers have seen a gr...
متن کاملOverpseudoprimes, Mersenne Numbers and Wieferich Primes
We introduce a new class of pseudoprimes-so called ”overpseudoprimes” which is a special subclass of super-Poulet pseudoprimes. Denoting via h(n) the multiplicative order of 2 modulo n,we show that odd number n is overpseudoprime if and only if the value of h(n) is invariant of all divisors d > 1 of n. In particular, we prove that all composite Mersenne numbers 2 − 1, where p is prime, and squa...
متن کاملMersenne 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 ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- SIAM J. Discrete Math.
دوره 25 شماره
صفحات -
تاریخ انتشار 2011