Big Numbers, NOT Big Data
Continued from my post in X (Twitter):
Around 10+ years ago, Big Data became very popular in IT industry, which I thought would accelerate AI. It also brought a new word "Data Scientist", which I am far from being, but the sound of that word "scientist" has remained in my memory in conjunction with Kenji Miyazawa 宮沢賢治 (1896-1933), who responded to Shimpei Kusano 草野心平 (a Japanese poet, 1903-1988) in a letter in April 1924 (100 years ago) as
"I am not sure about being a poet, but I would like to be recognized as a scientist."While I was not engaged in any particular business using Big Data analysis, I studied both applications as well as such technologies at superficial level.
However, it caused a different interest to arise --- Big Numbers, NOT Big Data.
Just picking up several examples, Googolplex, Leviathan Number, Legion's Numbers, Goliath Numbers, Skewes' Number, Graham's Number, Busy Beaver Numbers, Rayo's Number…
It might not be obvious for those who don't have any background in elementary mathematics, factorization of large numbers such as decimal digit > 100 (for example) is extremely time consuming, which is effectively the backbone theory that sustains cryptography. The elliptic curves on finite fields is a beautiful theory in mathematics and has been applied to cryptography extensively, but let's not dig into this particular topic.
Looking at GIMPS (Great Internet Mersenne Prime Search), it's so obvious that non-trivial efforts have been made to find large size prime numbers.
https://en.wikipedia.org/wiki/Great_Internet_Mersenne_Prime_Search
As briefly mentioned in my article on मंत्र (mantra), I love the following three words:
恒河沙 gōgasha (coined from Sanskrit word गङ्ग which means the Ganges),
阿僧祇 asōgi (transliterated from Sanskrit असंख्येय), and
那由他 nayuta (borrowed from Sanskrit नयुत)
According to Wikipedia, the word 恒河沙 had been already used in the Heian Period (794–1185) as mentioned in Konjaku Monogatarishū 今昔物語集,
那由他 & 阿僧祇 were used like 「五百四十万億那由他劫、大通智勝仏の寿命は五百四十万億那由他劫」and 「無量千万億の阿僧祇の世界」respectively by 法華経 Lotus Sutra सद्धर्मपुण्डरीक सूत्र.
Here another letter 劫 (kō) appears with a sample in 法華経 Lotus Sutra being used in a word 塵点劫 (jintengō), which means indescribably long period of time.
It was used as a unit of time and means "a long period of time (aeon) related to the lifetime of the universe (creation)," according to Wikipedia, and the letter comes of 「劫波（劫簸）」, a transliteration of Sanskrit कल्प kalpa. A kalpa is equal to 4.32 billion years in Hinduism. This leads to another very interesting topic, which I hopefully will visit in future...
劫 is used in Jugemu 寿限無, a famous rakugo story in Japanese traditional entertainment, as "Gokō-no Surikire" 五劫の擦り切れ which means the following:
"five kō of rubbing off (the rock)". In Japanese Buddhist lore, a heavenly maiden would visit the human world once in every three thousand years, leaving friction marks on a huge rock with her dress. Eventually, the rock would wear down to nothing in the span of one kō, or 4 billion (4×10^9) years.
So 劫 kalpa कल्प is a complete opposite of Planck time 5.391247 × 10^(-44) seconds discovered by Max Planck at the end of the 19th century.
The letter 劫 was used in a title of a book on mathematics written by Yoshida Mitsuyoshi (吉田光由 1598-1673) in 1627, the early Edo era, which contains descriptions of 恒河沙, 阿僧祇 & 那由他 at the beginning:
This book is called Jinkōki 塵劫記, though it is not purely purposed to talk about such a topic of "Big Numbers" which is utterly useless for daily life but "contains instructions for dividing and multiplying with a soroban 算盤, an abacus developed in Japan, and mathematical problems relevant to merchants and craftsmen" as per Wikipedia.
The table of the contents (for part 1 & 2) are as follows:
This is just an excerpt from the book available at Iwanami Bunko (Iwanami paperback library) 岩波文庫:
And here is a copy of Revised Jinkōki (改算塵劫記) owned by the National Museum of Nature and Science 国立科学博物館 via Wikipedia:
One thing worth mentioning is that 3.16 is used instead of 3.14 as the circumference ratio π in this book, and was never corrected in the subsequent versions throughout Edo era, as shown in the following paragraph:
Let us recall that Seki Takakazu (関孝和), a Japanese mathematician (1642-1708), obtained a value for π that was correct to the 10th decimal place using a regular polygon with 131072 sides around 1681, and later his student Takebe Kenkō (建部賢弘) calculated 41 digits of π based on polygon approximation and Richardson extrapolation. Such accomplishment was never reflected to the bestseller Jinkōki unfortunately…
It's rather interesting that there is a monument inscribed with the words Jinkōki in Kyōto, which comes from that the author of Jinkōki was born in the area where Jōjakkōji Temple is located.
I had an opportunity to visit Jōjakkōji Temple 常寂光寺 in Ogura-yama, Saga, Ukyō-ku, Kyōto on 22nd May 2024. It was a very sunny day. I walked to the temple from Saga-Arashiyama Station (嵯峨嵐山駅) of JR San'in Main Line (JR山陰本線). As expected I saw more foreigners than Japanese, but that's a different story.
Back to gōgasha 恒河沙, asōgi 阿僧祇 and nayuta 那由他, they are nothing compared with "Gigaprimes", yet for our daily life they are still absurdly big and none would encounter a situation that requires calculation at such level.(The record of the largest prime is currently held by Mersenne prime 2^82,589,933 − 1 with 24,862,048 digits, found by GIMPS in December 2018. A Gigaprime is a prime number whose decimal representation has 1,000,000,000 or more digits.)
There is an open problem in mathematics called "Sums of three cubes":
Is there a number that is not 4 or 5 modulo 9 and that cannot be expressed as a sum of three cubes?
In 2019 a solution was discovered by Andrew Booker for the case n = 33.
https://www.newsweek.com/uncracked-problem-mathematician-diophantine-puzzle-1384422
An American mathematician has cracked part of a problem that had remained unsolved for 64 years.
Andrew Booker, Reader of Pure Mathematics at the University of Bristol in the U.K., worked out how to express the number 33 as the sum of three cubes. (8,866,128,975,287,528)³ + (–8,778,405,442,862,239)³ + (–2,736,111,468,807,040)³.
A Wikipedia entry for nayuta 那由他 contains the case for n = 3:
and describes each entry using gōgasha 恒河沙, asōgi 阿僧祇, and nayuta 那由他:
185那由他1314阿僧祇2647恒河沙0358極7210載3000正3064澗5504溝8912穣0286𥝱0631垓5008京9838兆9977億4924万8000
-185那由他1314阿僧祇2636恒河沙4725極7462載8907正3278澗1685溝4239穣9539𥝱6198垓0212京7338兆9089億4467万1229
-10恒河沙5632極9747載4092正9786澗3819溝4672穣0746𥝱4433垓4796京2500兆0888億0457万6768
…and this is probably the first and the last opportunity for me to describe actual numbers using gōgasha 恒河沙, asōgi 阿僧祇, and nayuta 那由他 □
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