No, Graham’s number is not a recursive number.
Detailed answer question
Graham’s number is a very large number created by Ronald Graham. It is considered one of the largest numbers to have ever been used in a mathematical proof. It was first described in 1971, as part of a proof relating to the Ramsey theory.
Graham’s number is so large that it is practically impossible to write down in its entirety. It is a number far too large for any practical purposes, and is mainly used to demonstrate the concept of really, really big numbers.
The number is illustrated using tetration, a mathematical operation that is essentially repeated exponentiation. It is far beyond the comprehension of most people, however, it is not a recursive number.
In a quote from Ronald Graham himself, he said “I believe that Graham’s number is originally mainly considered as a joke, allowing you to prove a certain statement which is so ridiculous that nobody even would have conjectured cause you can’t even write the number down”.
Below is a table of some interesting comparisons to demonstrate just how unimaginably enormous Graham’s number is.
| Comparison | Number |
|---|---|
| Number of atoms in the observable universe | 10^80 |
| Skewes’ number (a very large non-Graham’s number) | 10^10^10^34 |
| Graham’s Number | 3 ↑↑↑↑ 3 |
| Number of particles in the observable universe | 10^86 |
| Number of Planck volumes in the observable universe | 10^185 |
In conclusion, Graham’s number is not a recursive number. It is an extremely large number that is used more for demonstrating how truly enormous numbers can get than for any practical purpose.
Video answer to “Is Graham’s number a recursive number?”
The video discusses two number sequences: the TREE(n) and g(n), which can be extended into unimaginably large numbers, and the comparison between the growth rates of the two sequences. The speaker introduces the concept of fast-growing hierarchies, using sequences like successor, multiplication, exponentiation, and tetration, to measure the growth rates of the sequences. They then introduce a function, f_omega, which grows more quickly than any function that came before it and explain how to define ordinal numbers using Omega plus 1, Omega plus 2, and so on until reaching Omega times 3. The hierarchy includes Epsilon, Eta, and Veblen hierarchies, leading up to the Feferman-Schütte ordinal, gamma 0. The function f(gamma0) grows incredibly fast but cannot keep pace with the TREE sequence, which is even faster. The speaker concludes that TREE of Graham’s number is larger than Graham of TREE.
There are alternative points of view
We can express the number of digits Graham’s number has using various recursive sequences or simply as a base 10 logarithm of Graham’s number itself, but it is far too large for us to be able to imagine or conceive in its full glory and magnitude, let alone being able to write it down somewhere.
An answer was proved to exist by Graham and Rothschild (1971), who also provided the best known upper bound, given by (1) where Graham’s number is recursively defined by (2) and (3) Here, is the so-called Knuth up-arrow notation. is often cited as the largest number that has ever been put to practical use (Exoo 2003).
number of subarrays whose minimum element is a specific given number
Number of pairs of numbers whose product is less than a given number
While Justin’s answer is absolutely correct, Graham’s Number and the fast-growing hierarchy of functions are not directly comparable, there is a sensible answer to this question:
That is, the first function in the fast growing hierarchy that surpases Graham’s Number for a concise input. Specifically [math]f_{\omega+1}(64) %3E [/math] Graham’s number. There’s a bit more explanation in the Wikipedia article: Fast-growing hierarchy [ http://en.wikipedia.org/wiki/Fast-growing_hierarchy#Functions_in_fast-growing_hierarchies ].
But the basic explanation is that [math]f_\omega[/math] is approximately the Ackermann function and thus [math]f_{\omega + 1}[/math] is the iterated Ackermann function. Graham’s Number is defined as 64 iterations of the Ackermann function, so that function concisely dominates it.
I’m sure you will be interested
Is Graham’s number a real number?
Graham’s number (G) is a very big natural number that was defined by a man named Ronald Graham. Graham was solving a problem in an area of mathematics called Ramsey theory. He proved that the answer to his problem was smaller than Graham’s number.
How many 3s are in Graham’s number?
As a response to this: 7.6 trillion 3s
So it’s a power tower that has 3 3 3 occurrences of in it, or about 7.6 trillion 3s. And there you have it. We just broke our brain.
Why does Graham’s number end in a 7?
As an answer to this: Graham’s number is bigger than the googolplex. It’s so big, the Universe does not contain enough stuff on which to write its digits: it’s literally too big to write. But this number is finite, it’s also an whole number, and despite it being so mind-bogglingly huge we know it is divisible by 3 and ends in a 7.
Similar
What would happen if you memorize Graham’s number?
The reply will be: “If you even try to imagine Graham’s number, your head would break down into a black hole because your head cannot store the information required to envisage it,” website Physics Astronomy explains. The Graham’s number is so long that there is not enough space in the universe to write down all the digits.
Is Graham’s number computable?
However, Graham’s number can be explicitly given by computable recursive formulas using Knuth’s up-arrow notation or equivalent, as was done by Graham. As there is a recursive formula to define it, it is much smaller than typical busy beaver numbers.
What is the upper bound of Graham’s number?
Graham’s number, G, G, is much larger than N: N: {f^ {64} (4)}, f 64(4), where { f (n)\;=\;3\uparrow ^ {n}3}. f (n) = 3 ↑n 3. This weaker upper bound for the problem, attributed to an unpublished work of Graham, was eventually published and named by Martin Gardner in Scientific American in November 1977.
What is Graham’s number in chained arrow notation?
Response to this: where Graham’s number is recursively defined by Here, is the so-called Knuth up-arrow notation. is often cited as the largest number that has ever been put to practical use (Exoo 2003). In chained arrow notation, satisfies the inequality
What is the large number named after Ronald Graham?
Answer will be: This article is about the large number named after Ronald Graham. For the investing term named after Benjamin Graham, see Graham number. Graham’s number is an immense number that arose as an upper bound on the answer of a problem in the mathematical field of Ramsey theory.