In this paper, we introduce Mersenne and Mersenne-Lucas hybrinomial quaternions present some of their properties. Some identities are derived for these polynomials. Furthermore, give the Binet formulas, Catalan, Cassini, d’Ocagne identity generating exponential function quaternions.

I provide the details of the factorization of the Mersenne number 21061 − 1 by the Special Number Field Sieve. Although this factorization is easier than the completed factorization of RSA-768, it represents a new milestone for factorization using publicly available software.

We exhibit twelve new primitive trinomials over GF(2) of record degrees 42 643 801, 43 112 609, and 74 207 281. In addition we report the first Mersenne exponent not ruled out by Swan’s theorem [10] — namely 57 885 161 — for which none primitive trinomial exists. This completes the search for the currently known Mersenne prime exponents. Primitive trinomials of degree up to 32 582 657 were repo...

All primitive trinomials over GF (2) with degree 859433 (which is the 33rd Mersenne exponent) are presented. They are X859433 +X288477 + 1 and its reciprocal. Also two examples of primitive pentanomials over GF (2) with degree 86243 (which is the 28th Mersenne exponent) are presented. The sieve used is briefly described.

We describe how we reached a new factoring milestone by completing the first special number field sieve factorization of a number having more than 1024 bits, namely the Mersenne number 2 − 1. Although this factorization is orders of magnitude ‘easier’ than a factorization of a 1024-bit RSA modulus is believed to be, the methods we used to obtain our result shed new light on the feasibility of t...

The Mersenne Twister is a 623-dimensionally equidistributed variant of the twisted generalized feedback shift register operating in 623 dimensions [6]. It is quite fast and produces a sequence of pseudo-random numbers with an enormous period and uniform distribution, which makes it suitable for use in simulations. However, the Mersenne twister is not suitable for cryptographic use: by analyzing...