Binary operations applied to numbers

1A binary operation is a calculus that combines two elements to obtain another elements. It seems quite simple for numbers, because we usually imagine it as a simple sum or product. However, also in the case of numbers, a binary operation can be extremely fascinating if we consider it in a generalized form. Here the reader can find several examples of generalized sums for different sets of numbers (Fibonacci, Mersenne, Fermat, q-numbers, repunits and many other numbers). These sets can form groupoid which possess different binary operators. As we will see at the end of this exposition of cases, the most relevant finding is that different integer sequences can have the same binary operator and that, consequently, can be used as different representations of the same groupoid.openopenAmelia Carolina SparavignaSparavigna, Amelia Carolin

A primality test for Kp^n+1 numbers

In this paper we generalize the classical Proth's theorem and the Miller-Rabin test for integers of the form N = Kpn +1. For these families, we present variations on the classical Pocklington's results and, in particular, a primality test whose computational complexity is Õ(log2 N) and, what is more important, that requires only one modular exponentiation modulo N similar to that of Fermat's test

A primality test for Kp^n+1 numbers

In this paper we generalize the classical Proth's theorem and the Miller-Rabin test for integers of the form N = Kpn +1. For these families, we present variations on the classical Pocklington's results and, in particular, a primality test whose computational complexity is Õ(log2 N) and, what is more important, that requires only one modular exponentiation modulo N similar to that of Fermat's test

Fibonacci s-Cullen and s-Woodall Numbers

Abstract The m-th Cullen number C m is a number of the form m2 m + 1 and the m-th Woodall number W m has the form m2 m − 1. In 2003, Luca and Stȃnicȃ proved that the largest Fibonacci number in the Cullen sequence is F 4 = 3 and that F 1 = F 2 = 1 are the largest Fibonacci numbers in the Woodall sequence. Very recently, the second author proved that, for any given s > 1, the equation F n = ms m ± 1 has only finitely many solutions, and they are effectively computable. In this note, we shall provide the explicit form of the possible solutions

Reflections on the number field sieve

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