SHF:Small:Arithmetic Algorithms and Applications of Hereditarily Binary Numbers

数字表示法已经从一元表示法（墙上的一个划痕代表一个单位）发展到以n为基数的数字系统，具有对数表示法大小的显著优势。在过去的1000年里，这种表示法被证明具有不同寻常的弹性，部分原因是所有实际计算都可以在这种表示法中以合理的效率进行。不过，有人可能会问，完全不同的编号系统能否提供类似的服务。更进一步，有人可能想问，是否有数字系统可以容纳更大的数字，同时仍然提供与熟悉的数字相当有效的计算。这个项目有几个实际用途，包括对非常大的数字进行有效的计算，这对数论和密码学等领域很重要。该项目研究的算法工作与这样的编号系统：遗传二进制数。它涉及到通过递归压缩出现在数字二进制表示中的0和1序列获得的树的计算。有了它们，计算巨大的数字，包括指数塔和所有记录质数变得容易处理。该项目包括使用集合和布尔逻辑进行紧凑型表示和计算，研究与一系列组合对象的连接，先进的数论算法以及实现遗传二进制数及其应用计算的开源软件库。

Number representations have evolved from the unary representation where one scratch on the wall represented a unit, to the base-n number system, with the remarkable benefit of a logarithmic representation size. Over the last 1000 years, this representation has proved to be unusually resilient, partly because all practical computations could be performed with reasonable efficiency within the notation. Still, one might ask if radically different numbering systems can offer similar services. And one step further one might want to ask if there are number systems that can accommodate significantly larger numbers while still providing comparably efficient computations with the familiar ones. This project has several practical uses involving efficient computations with very large numbers which is important for fields like number theory and cryptography.The project studies algorithms working with such a numbering system: hereditarily binary numbers. It involves computing with trees obtained by recursively compressing sequences of zeros and ones occurring in the binary representation of a number. With them, computations with giant numbers, including towers of exponents and all record holding prime numbers become tractable. The project includes work on compact representation and computation with sets and boolean logic, study of connections to a family of combinatorial objects, advanced number-theoretical algorithms as well as an open source software library implementing computations with hereditarily binary numbers and their applications.