Googol

A googol is a large number equal to  (i.e., a 1 with 100 zeros following it). Written out explicitly,

10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.

The term was coined in 1938 by 9-year-old Milton Sirotta, nephew of Edward Kasner (Kasner 1989, pp. 20-27; Bialik 2004). Kasner then extended the term to the larger "googolplex." It should be noted that "googol" is indeed the correct spelling of the term, so the spelling "Google" refers to the internet search engine, not one with 100 zeros.

The residues of googol (mod ) for , 2, ... are 0, 0, 1, 0, 0, 4, 4, 0, 1, 0, ... (OEIS A066298).

The integer sequence  that counts the number of  matrices over an alphabet of size 10 and having first few terms 10, 10000, 1000000000, ... (OEIS A076782) reaches googol at its 10th term, and googolplex () at its  term. Since  grows exponentially, this gives an idea of how big a googolplex is with respect to the size of googol.

REFERENCES:

Bialik, C. "Searching for the Birth of the Googol." The Wall Street J. Jun. 19, 2004. https://afr.com/articles/2004/06/18/1087245103935.html.

Flannery, S. and Flannery, D. In Code: A Mathematical Journey. London: Profile Books, pp. 112-113, 2000.

Kasner, E. and Newman, J. R. Mathematics and the Imagination. Redmond, WA: Tempus Books, pp. 20-27, 1989.

Pappas, T. "Googol & Googolplex." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 76, 1989.

Sloane, N. J. A. Sequences A066298 and A076782 in "The On-Line Encyclopedia of Integer Sequences."