Relative Goldbach Partitions and Goldbach\u27s Conjecture

In this note, we utilize techniques from discrete mathematics to develop first an inequality, and then second a counting formula that is connected to Goldbach\u27s conjecture. In order to do this, we introduce the notion of a Relative Goldbach Partition

Relative Goldbach Partitions and Goldbach\u27s Conjecture

In this note, we utilize techniques from discrete mathematics to develop first an inequality, and then second a counting formula that is connected to Goldbach\u27s conjecture. In order to do this, we introduce the notion of a Relative Goldbach Partition

Cyclone Codes

We introduce Cyclone codes which are rateless erasure resilient codes. They combine Pair codes with Luby Transform (LT) codes by computing a code symbol from a random set of data symbols using bitwise XOR and cyclic shift operations. The number of data symbols is chosen according to the Robust Soliton distribution. XOR and cyclic shift operations establish a unitary commutative ring if data symbols have a length of $p-1$ bits, for some prime number $p$. We consider the graph given by code symbols combining two data symbols. If $n/2$ such random pairs are given for $n$ data symbols, then a giant component appears, which can be resolved in linear time. We can extend Cyclone codes to data symbols of arbitrary even length, provided the Goldbach conjecture holds. Applying results for this giant component, it follows that Cyclone codes have the same encoding and decoding time complexity as LT codes, while the overhead is upper-bounded by those of LT codes. Simulations indicate that Cyclone codes significantly decreases the overhead of extra coding symbols

10 conjectures in additive number theory

Following an idea of Rowland we give a conjectural way to generate increasing sequences of primes using algorithms involving the gcd. These algorithms seem not so useless for searching primes since it appears we found sometime primes much more greater than the number of required iterations. In an other hand we propose new formulations of famous conjectures from the additive theory of numbers (the weak twin prime conjecture, the Polignac conjecture, the Goldbach conjecture or the very general Schinzel's hypothesis H). For the moment these are experimental results obtained using pari-gp.Comment: 45 pages, 18 figure