遗传二进制自然数的算术算法
Arithmetic Algorithms for Hereditarily Binary Natural Numbers
摘要：
我们研究了一个新的基于树的数字表示，{ \em遗传二进制数}，通过应用双射基2数字的递归行程长度编码定义的一些基本的算术性质。我们的表示法将已知的最大素数及其相关的完全数以及已知的最大Woodall、Cullen、Proth、Sophie Germain和孪生素数等巨数表示为小尺寸的树。更重要的是，我们的数字表示支持新的算法，在最好的情况下，崩溃的各种计算的复杂性的超指数因子，并在更坏的情况下是在一个恒定的因素，他们的传统同行。因此，它打开了一扇通往新世界的大门，在这个世界中，算术运算受到其操作数的结构复杂性的限制，而不是它们的位大小。
英文摘要：
We study some essential arithmetic properties of a new tree-based number representation, {\em hereditarily binary numbers}, defined by applying recursively run-length encoding of bijective base-2 digits. Our representation expresses giant numbers like the largest known prime number and its related perfect number as well as the largest known Woodall, Cullen, Proth, Sophie Germain and twin primes as trees of small sizes. More importantly, our number representation supports novel algorithms that, in the best case, collapse the complexity of various computations by super-exponential factors and in the worse case are within a constant factor of their traditional counterparts. As a result, it opens the door to a new world, where arithmetic operations are limited by the structural complexity of their operands, rather than their bitsizes.
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