What is the difference between graham’s number and skewes’ number?

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Graham’s number relates to a specific mathematical problem, while Skewes’ number is related to the distribution of primes.

So let us examine the query more closely

Graham’s number and Skewes’ number are both significant in the field of mathematics, but they pertain to different areas. Graham’s number was first described by mathematician Ronald Graham in 1971 and is an extremely large number that was used to solve a problem in the field of Ramsey theory. According to Graham himself, the number is so large that even writing every digit in the observable universe would not be enough to express it.

On the other hand, Skewes’ number is related to the distribution of prime numbers. It was discovered by mathematician Stanley Skewes in 1933 when he was investigating Bertrand’s postulate, which states that there is always a prime number between n and 2n. Skewes’ number is the smallest number for which the actual number of primes up to that point is greater than the number predicted by the Riemann hypothesis. This number is lower than Graham’s number, but it is still an incredibly large number.

In order to truly understand the magnitude of these numbers, it may be helpful to compare them to other well-known values. For example, Graham’s number is so large that it’s estimated to be much larger than the number of atoms in the observable universe. Skewes’ number is also staggeringly large – it was once cited as the largest number ever used in a published mathematical proof.

To further illustrate the difference between these numbers, a table is provided below that highlights some key characteristics:

Graham’s number Skewes’ number
First discovered 1971 1933
Related to Ramsey theory Distribution of prime numbers
Significance Extremely large, difficult to comprehend Used in published mathematical proof
Magnitude Estimated to be incredibly large, larger than atoms in observable universe Also very large, once the largest number used in published proof
Calculation method Recursive Calculated using the Riemann hypothesis
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In summary, while both Graham’s number and Skewes’ number are substantial numbers, they represent different concepts in mathematics. Graham’s number is related to Ramsey theory and is incredibly large, while Skewes’ number is related to the distribution of prime numbers and is also very large, once holding the title of largest number used in a proof.

As mathematician Terence Tao puts it, “these numbers are jokes, basically – they’re just saying, it’s so large that you can’t say anything meaningful about it.”

See a video about the subject.

Mathematicians Tony Padilla and Matt Parker discuss arrow notation, used to represent very large numbers and particularly in combinatorics problems. They discuss the concept of Graham’s number and its development as the maximum possible number of people needed to be in committees with certain conditions on connections, using arrow notation to show how the number increases and its scale, which lies between 6 and Graham’s number. Despite being smaller than infinity, which is currently used in mathematical proofs, Graham’s number is shockingly large, with only its last 500 digits known and its first digit unknown. The video ends with an interesting anecdote about Graham, who was a mathematician and circus performer.

See further online responses

Graham’s number is much larger than many other larger numbers such as Skewes’ number and Moser’s Number, both of which are in turn much larger than a googolplex.

Skewes was particularly interested in prime numbers, and, when his number was introduced in 1933, it was described by a colleague as "largest number which has ever served any definite purpose in mathematics." However, Skewes’ number has since lost that distinction to Graham’s number, which is currently designated as the world’s largest number.

Graham’s number is a mind-blowingly large number which is so unimaginably large, that you have never even tried thinking of numbers of this magnitude until you learned what it exactly means!

But Graham’s number has a strict mathematical definition. To understand it, you need to understand how mathematical operators work.

If you are ready to understand something totally new, then let’s go:

1. The very basic mathematical operator which increases an original number is successor. Speaking simple words, it is just increasing a positive integer number by “1”: Succ (1) = 2; Succ (2) = 3; Succ (3) = 4; Succ (16) = 17; Succ (7 128) = 7 129 etc. The idea is pretty clear – OK, let’s call it rank-0 mathematical operation.
2. If repeating successor of a number (a) multiple times (b), we get the next operation, which is called addition. To understand how exactly addition works, let’s consider an example: 7 + 3 = 10. Here “3” shows how many times successor is repeated, literally it reads the follo…

What is higher than Skewes?
Response to this: However, Skewes’ number is no longer considered the largest possible number; that title now goes to Graham’s number. Graham’s number, which can’t be written with conventional notation, was developed by mathematician R.L. Graham.
What does Skewes number represent?
The Skewes’ numbers are large upper-bounds to the solution of a problem whose answer is still not known, and they were named after Stanley Skewes who proved them to be upper-bounds.
Is there a number bigger than Graham's?
Answer will be: Now, there are larger numbers used in mathematics which have dethroned Graham, such as TREE(3), SCG(13), and other terms in far-reaching sequences. Nevertheless, Graham’s number remains famous, and is still mistakenly thought by many as "the largest useful number".
How many zeros are behind Graham's number?
It is a one followed by 100 zeros. (Fun fact: this number inspired the name of the search engine Google, but the company’s founders accidentally misspelled it when checking whether the web domain was still available. The rest is history.) To get much bigger than that, we need to put larger numbers into the exponent.
What is Graham's number?
Response will be: Grahams number is an immense number that arose as an upper bound on the answer of a problem in the mathematical field of Ramsey theory. It is much larger than many other large numbers such as Skewess number and Moser’s number, both of which are in turn much larger than a googolplex.
Did Skewes prove that there is a first number?
Skewes’s numbers J.E. Littlewood, who was Skewes’sresearch supervisor, had provedin Littlewood (1914)that there is such a number (and so, a first such number); and indeed found that the sign of the difference π(x)−li(x){\\displaystyle \\pi (x)-\\operatorname {li} (x)}changes infinitely many times.
Why is the Skewes number so large?
The reason why the Skewes number is so large is that these smaller terms are quite a lotsmaller than the leading error term, mainly because the first complexzero of the zeta function has quite a large imaginary part, so a large number (several hundred) of them need to have roughly the same argument in order to overwhelm the dominant term.
Why is Graham's number a power tower?
Answer to this: Graham’s number is a "power tower" of the form 3↑↑ n (with a very large value of n ), so its rightmost decimal digits must satisfy certain properties common to all such towers. One of these properties is that all such towers of height greater than d (say), have the same sequence of d rightmost decimal digits.

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