On the Arithmetic of Knuth’s Powers and Some Computational Results About Their Density

Abstract

The object of the paper are the so-called "unimaginable numbers". In particular, we deal with some arithmetic and computational aspects of the Knuth's powers notation and move some first steps into the investigation of their density. Many authors adopt the convention that unimaginable numbers start immediately after 1 googol which is equal to 10 100 , and G.R. Blakley and I. Borosh have calculated that there are exactly 58 integers between 1 and 1 googol having a nontrivial "kratic representation", i.e., are expressible nontrivially as Knuth's powers. In this paper we extend their computations obtaining, for example, that there are exactly 2 893 numbers smaller than 10 10 000 with a nontrivial kratic representation, and we, moreover, investigate the behavior of some functions, called krata, obtained by fixing at most two arguments in the Knuth's power a ↑ b c.

Some Variants of Integer Multiplication

“…But now, if t → 0 − , then c 2 → +∞. Hence, the positive semi-axis represented in (5) coincides with the usual one, and the result follows.…”

“…With this result, we justify, in a certain way, Wallis's paradox about negative numbers ( [7], p. 253): "In his Arithmetica Infinitorum (1655), he argued that since the ratio a/0, when a is positive, is infinite, then, when the denominator is changed to a negative number, as in a/b with b negative, the ratio must be greater than infinity". However, we would obtain other paradoxes, such as the appearance of the "transposed infinity" in (5) and the interpretation of Hilbert's hotel [8] in this version of the number line.…”

“…This manuscript deals with typical themes of the Pythagorean mathematical heritage. We recommend to the interested reader in this topic to investigate the bibliography contained in [5,6].…”

Beyond Knuth's notation for unimaginable numbers within computational number theory

“…For new density results on Knuth's powers see [12], or see also [10,11] for some links between unimaginable numbers, gross-one and new algebraic and geometric constructs arising from Fibonacci numbers.…”