We know many largest numbers as googol number(=10^100), googolplex number(=10^googol), and other unimaginable large numbers.
What is the largest known number that is the result of solving a problem in physics or mathematics?
Are they really practical or just based on conjecture?
Dear Jamshidi
Some Coefficients of the chemical and physical equations(like as Avogadro constant:NA = 6.02214076×10²³ ) are big. But I mean the largest known number that comes from solving a real problem.
Regards
Hello Mr Seyyed Hossein Jafari Petroudi
In the answer to your question, I can say that I have seen very few such values existing in this world. They are as follows:
That's all I could see in my life. I hope that will help you. Good Luck!
Dear Sahu
Your answer was useful. As I said to Dr Jamshidi, some coefficients of the chemical and physical equations are big. I mean the largest known number that is obtained by solving a problem.
Regards
I think you mean algebraic numbers; An infinite set. Are you meaning the entitled or named numbers.
Dear Jamshidi
You are right. It can be the solution of a (well known) algebraic equation.
Regards
The largest such numbers I know of come from combinatorial problems, e.g., Ramsey Theory
In the search for prime numbers, very large numbers have been obtained,
the last one being perhaps 2⁸² ⁵⁸⁹ ⁹³³ - 1. Thomas I. Seidman recalled Ramsey theory and combinatorial problems, that require even larger numbers.
If your interest is not limited to finite quantities, there are the transfinite numbers: aleph_0 is the smallest and it is larger than every finite number; about their use, aleph_0 and c (the cardinality of R) appear in measure theory and some related problems.
There are an infinite number of primes. Take the largest know prime number, then there is a prime number larger. Then you have the cardinals. Let X be a set the cardinality of X is the number of elements. For X={1,2}, the carnality is of course 2. However, one very quickly runs out of finite numbers which leads into different sizes of infinity. For example the carnality of the natural number is Aleph null, the smallest infinite cardinal. The carnality of the real numbers is the cardinal "c" and c>Aleph null. Some text refer to c by Aleph One. For any set the power set can be formed - the set of all subsets. The carnality of the power set is 2 raised to the power of the carnality of the original set.
The Continuum Hypothesis proposed by Cantor in 1878 and listed as the first of Hilbert's 23 problems in 1900 states that 2^Aleph null = c, that is there is no cardinal number between Aleph null and c.
In mathematics - the question is somewhat irrelevant. In physics, maybe not depending but then again...
About Truman Prevatt' answer: Of course I was speaking about _verified_ primes. If you like transfinite numbers, there are inaccessible, Mahlo and hyper-Mahlo transfinite cardinal numbers, too.
Found also: Skewes's number. It needed initially very large upper bounds. See https://en.wikipedia.org/wiki/Skewes%27s_number and Bibliography.
Some of the numbers from Ramsey theory, such as Graham’s number, have been mentioned already. A number from combinatorics that is far bigger is Tree(3). Much bigger numbers can be found in the theory of computation. One example is Loader’s number, which is the largest number that can be computed by a C program in at most 512 characters. The busy beaver numbers come from the problem of the largest possible halting time of a Turing machine with a program of at most n lines (I recommend the Quanta article www.quantamagazine.org/the-busy-beaver-game-illuminates-the-fundamental-limits-of-math-20201210/). This was used by Rayo to define Rayo’s number, which is the largest number expressible in first-order logic using at most a google symbols – this allowed him to win the ‘big number duel’. They can be made even larger by adding variations such as ‘super Turing machines’ and halting oracles. A caveat is that these larger numbers coming from busy beavers are in general not computable. Some larger numbers that have been proposed, such as BIG FOOT, have been found to be ill-defined. Around the middle of this answer we have transitioned from ‘a big number that solves a genuine mathematical problem’ to ‘finding a big number for the sake of it’, so I suggest some of the busy beaver numbers as an answer to the question. There are some great Numberphile videos on Youtube on all of the subjects that I have mentioned.
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